Regions of linearity, Lusztig cones and canonical basis elements for the quantized enveloping algebra of type A_{4} ^{1}

Roger Carter

Mathematics Institute, University of Warwick, Coventry CV4 7AL, England E-mail: rwc@maths.warwick.ac.uk

Robert Marsh

Department of Mathematics and Computer Science, University of Leicester, University Road, Leicester LE1 7RH, England E-mail: R.Marsh@mcs.le.ac.uk

To Professor Helmut Wielandt on his 90th birthday.

Abstract

Let U_{q} be the quantum group associated to a Lie algebra g of rank n. The negative part U^{−} of U has a canonical basis B with favourable properties (see Kashiwara [6] and Lusztig [11, §14.4.6]). The approaches of Lusztig and Kashiwara lead to a set of alternative parametrizations of the canonical basis, one for each reduced expression for the longest word in the Weyl group of g. We show that if g is of type A_{4} there are close relationships between the Lusztig cones, canonical basis elements and the regions of linearity of reparametrization functions arising from the above parametrizations. A graph can be defined on the set of simplicial regions of linearity with respect to adjacency, and we further show that this graph is isomorphic to the graph with vertices given by the reduced expressions of the longest word of the Weyl group modulo commutation and edges given by long braid relations. Keywords: Quantum group, Lie algebra, Canonical basis, Tight monomials, Weyl group, Piecewise-linear functions.

1 Introduction

Let U = U_{q}(g) be the quantum group associated to a semisimple Lie algebra g of rank n. The negative part U^{−} of U has a canonical basis B with favourable properties (see Kashiwara [6] and Lusztig [11, §14.4.6]). For example, via action on highest weight vectors it gives rise to bases for all the finitedimensional irreducible highest weight U-modules.

Let W be the Weyl group of g, with Coxeter generators s_{1}, s_{2}, …, s_{n}, and let w_{0} be the element of maximal length in W. Let i be a reduced expression for w_{0}, i.e. w_{0} = s_{i_{1}}s_{i_{2}} ⋯ s_{i_{k}} is reduced. Lusztig obtains a parametrization of the canonical basis B for each such reduced expression i, via a correspondence between a basis of PBW-type associated to i and the canonical basis. This gives a bijection

ϕ_{i} : B → ℕ^{k},

1Technical Report No. 2000/17, Department of Mathematics & Computer Science, University of Leicester, University Road, Leicester LE1 7RH, England.

where ℕ = {0, 1, 2, … }.

Kashiwara, in his approach to the canonical basis (which he calls the global crystal basis), defines certain root operators F̃_{i} on the canonical basis (see [6, §3.5]) which lead to a parametrization of the canonical basis for each reduced expression i by a certain subset Y_{i} of ℕ^{k}. This gives a bijection ψ_{i} : B → Y_{i}. The subset Y_{i} is called the string cone.

In Lusztig’s theory, an important role is played by two specific reduced expressions j and j′, which in type A_{4} with Dynkin diagram as in Figure 1 are j = (1, 3, 2, 4, 1, 3, 2, 4, 1, 3) and j′ = (2, 4, 1, 3, 2, 4, 1, 3, 2, 4).

Figure 1: Dynkin diagram of type A^{4}^{∘}  ^{∘}  ^{∘}  ^{∘}^{1}  ^{2}  ^{3}  ^{4}

The function R_{j}^{j′} = ϕ_{j′}ϕ_{j}^{−1} : ℕ^{k} → ℕ^{k} was shown by Lusztig to be piecewise-linear, and the regions of linearity of this function were shown to be relevant to understanding the behaviour of the canonical basis.

The function S_{i}^{j} = ϕ_{j}ψ_{i}^{−1} : Y_{i} → ℕ^{k} is useful in relating Kashiwara’s and Lusztig’s parametrizations of B. These re-parametrization functions have recently been studied using an approach involving totally positive varieties, in the preprint [4] of Berenstein and Zelevinsky (although not in terms of regions of linearity).

There is no simple way to express the elements of B in terms of the natural generators F_{1}, F_{2}, …, F_{n} of U^{−}. This has been done only in types A_{1}, A_{2}, A_{3} and B_{2} (see [8, §3.4], [12, §13], [20] and [22]) and appears to become arbitrarily complicated in general. However it seems that there is an interesting subset of B whose elements are expressible as monomials in F_{1}, F_{2}, …, F_{n}.

A monomial F_{i_{1}}^{(a_{1})}F_{i_{2}}^{(a_{2})} ⋯ F_{i_{k}}^{(a_{k})}, where F_{i}^{(a)} = F_{i}^{a}/[a]!, is said to be tight if it belongs to B. Lusztig [12] described a method which in low rank cases leads to the construction of tight monomials. He defined, for each reduced expression i of w_{0}, a certain cone C_{i} in ℕ^{k} which we shall call the Lusztig cone, and showed that, for types A_{1}, A_{2}, A_{3}, the above monomial is tight for all a ∈ C_{i}. In [14] the second author showed that this is also true in type A_{4}. Let

M_{i} = {F_{i_{1}}^{(a_{1})}F_{i_{2}}^{(a_{2})} ⋯ F_{i_{k}}^{(a_{k})} : a ∈ C_{i}}

be the set of monomials obtained from elements of C_{i}.

There appears to be an intriguing relationship between the Lusztig cones, the regions of linearity of Lusztig’s function R_{j}^{j′}, and the tight monomials in the canonical basis. This relationship is not fully understood in general, but the aim of this paper is to describe this relationship in the type A_{4} case. Types A_{1}, A_{2}, A_{3} were considered by Lusztig, but the A_{4} case is considerably more complicated, while at the same time being amenable to explicit calculation.

In type A_{4} we can show the following. For each a ∈ C_{i} we have

F̃_{i_{1}}^{a_{1}} F̃_{i_{2}}^{a_{2}} ⋯ F̃_{i_{k}}^{a_{k}} · 1 ≡ F_{i_{1}}^{(a_{1})}F_{i_{2}}^{(a_{2})} ⋯ F_{i_{k}}^{(a_{k})} mod vL′,

where L′ is the A-lattice spanned by B and A is the subring of ℚ(v) of functions regular at v = 0. Since it has been shown (independently by the second author [15] and Premat [17]) that C_{i} ⊆ Y_{i}, it follows that M_{i} ⊆ B and that, under the Kashiwara parametrization of B, we have

ψ_{i}(M_{i}) = C_{i}.

We can also show that, under Lusztig’s parametrization of B, we have

ϕ_{j}(M_{i}) = X_{i},

where X_{i} is one of the regions of linearity of the piecewise-linear function R_{j}^{j′} and that the transition function

S_{i}^{j} : C_{i} → X_{i}

is linear for all i.

The set of all regions of linearity of R_{j}^{j′} in type A_{4} was determined by the first author using the ideas outlined in [5]. These regions are described in the present paper. Each such region is defined by a certain set of inequalities. It turns out that the regions X_{i} are all defined by the minimal possible number of inequalities, and that they give the set of all regions defined by this minimal number of inequalities. We call these the simplicial regions of R_{j}^{j′}. We thus have a parametrization of the simplicial regions in terms of the reduced expressions i of w_{0} modulo commutation. A graph can be defined on the set of simplicial regions with respect to adjacency, and we further show that this graph is isomorphic to the graph with vertices given by the reduced expressions i of w_{0} modulo commutation and edges given by long braid relations.

Thus in type A_{4} there are close relationships between the Lusztig cones C_{i}, the tight monomials M_{i}, and the regions of linearity X_{i}. Examples of N. H. Xi [21] and M. Reineke [18] show, however, that these relationships cannot be expected to hold in the same way in type A_{n} for arbitrary n.

2 Parametrizations of the canonical basis

Let g be the simple Lie algebra over ℂ of type A_{n} and U be the quantized enveloping algebra of g. Then U is a ℚ(v)-algebra generated by the elements E_{i}, F_{i}, K_{μ}, i ∈ {1, 2, …, n}, μ ∈ Q, the root lattice of g. Let U^{+} be the subalgebra generated by the E_{i} and U^{−} the subalgebra generated by the F_{i}.

Let W be the Weyl group of g. It has a unique element w_{0} of maximal length. For each reduced expression i for w_{0} there are two parametrizations of the canonical basis B for U^{−}. The first arises from Lusztig’s approach to the canonical basis [11, §14.4.6], and the second arises from Kashiwara’s approach [6].

Lusztig’s Approach There is an ℚ-algebra automorphism of U which takes each E_{i} to F_{i}, F_{i} to E_{i}, K_{μ} to K_{−μ} and v to v^{−1}. We use this automorphism to transfer Lusztig’s definition of the canonical basis in [8, §3] to U^{−}.

Let T_{i}, i = 1, 2, …, n, be the automorphism of U as in [10, §1.3] given by:

T_{i}(E_{j}) = \{ \begin{matrix}−F_{j}K_{j}, & if i = j, \\ E_{j}, & if |i − j| > 1 \\ −E_{i}E_{j} + v^{−1}E_{j}E_{i} & if |i − j| = 1 \end{matrix}

T_{i}(F_{j}) = \{ \begin{matrix}−K_{j}^{−1}E_{j}, & if i = j, \\ F_{j}, & if |i − j| > 1 \\ −F_{j}F_{i} + vF_{i}F_{j} & if |i − j| = 1 \end{matrix}

T_{i}(K_{μ}) = K_{μ−⟨μ,α_{i}⟩h_{i}}, for μ ∈ Q,

where α_{i} are the simple roots and h_{i} the simple coroots of g.

For each i, let r_{i} be the automorphism of U which fixes E_{j} and F_{j} for j = i or |i − j| > 1 and fixes K_{μ} for all μ, and which takes E_{j} to −E_{j} and F_{j} to −F_{j} if |i − j| = 1. Let T′′_{i,−1} = T_{i}r_{i} be the automorphism of U as in [11, §37.1.3]. Let c ∈ ℕ^{k}, where k = ℓ(w_{0}), and i be a reduced expression for w_{0}. Let

F_{i}^{c} := F_{i_{1}}^{(c_{1})}T′′_{i_{1},−1}(F_{i_{2}}^{(c_{2})}) ⋯ T′′_{i_{1},−1}T′′_{i_{2},−1} ⋯ T′′_{i_{k−1},−1}(F_{i_{k}}^{(c_{k})}).

Define B_{i} = {F_{i}^{c} : c ∈ ℕ^{k}}. Then B_{i} is the basis of PBW-type corresponding to the reduced expression i. Let ¯ be the ℚ-algebra automorphism from U to U taking E_{i} to E_{i}, F_{i} to F_{i}, and K_{μ} to K_{−μ}, for each i ∈ [1, n] and μ ∈ Q, and v to v^{−1}. Lusztig proves the following result in [8, §§2.3, 3.2].

Theorem 2.1 (Lusztig) The Z[v]-span L of B_{i} is independent of i. Let π : L → L/vL be the natural projection. The image π(B_{i}) is also independent of i; we denote it by B. The restriction of π to L ∩ \overline{L} is an isomorphism of ℤ-modules π_{1} : L ∩ \overline{L} → L/vL. Also B = π_{1}^{−1}(B) is a ℚ(v)-basis of U^{−}, which is the canonical basis of U^{−}.

Lusztig’s theorem provides us with a parametrization of B, dependent on i. If b ∈ B, we write ϕ_{i}(b) = c, where c ∈ ℕ^{k} satisfies b ≡ F_{i}^{c} mod vL. Note that ϕ_{i} is a bijection.

Lusztig considers in [8] two particular reduced expressions for w_{0}. Let the nodes in the Dynkin diagram of A_{n} be labelled as in Figure 2.

Figure 2: Dynkin diagram of type A^{n}^{∘}  ^{∘}  ^{∘}  ^{∘}^{1}  ^{2}  ^{n − 1} ^{n}

Let j = 135 ⋯ 246 ⋯ 135 ⋯ 246 ⋯, and let j′ = 246 ⋯ 135 ⋯ 246 ⋯ 135 ⋯, where both expressions have length k = ℓ(w_{0}) — they are both reduced expressions for w_{0}. We have bijections ϕ_{j} : B → ℕ^{k} and ϕ_{j′} : B → ℕ^{k}.

Lusztig defines in [8, §2.6] a function R_{j}^{j′} = ϕ_{j′}ϕ_{j}^{−1} : ℕ^{k} → ℕ^{k}. This function was shown by Lusztig to be piecewise linear and its regions of linearity were shown to have significance for the canonical basis, in the sense that elements b of the canonical basis with ϕ_{j}(b) in the same region of linearity of R_{j}^{j′} often have similar form.

Kashiwara’s approach

Let Ẽ_{i} and F̃_{i} be the Kashiwara operators on U^{−} as defined in [6, §3.5]. Let A ⊆ ℚ(v) be the subring of elements regular at v = 0, and let L′ be the A-lattice spanned by arbitrary products F̃_{j_{1}} F̃_{j_{2}} ⋯ F̃_{j_{m}} · 1 in U^{−}. We denote the set of all such elements by S. The following results were proved by Kashiwara in [6].

Theorem 2.2 (Kashiwara)

(i) Let π′ : L′ → L′/vL′ be the natural projection, and let B′ = π′(S). Then B′ is a ℚ-basis of L′/vL′ (the crystal basis). (ii) Furthermore, Ẽ_{i} and F̃_{i} each preserve L′ and thus act on L′/vL′. They satisfy Ẽ_{i}(B′) ⊆ B′ ∪ {0} and F̃_{i}(B′) ⊆ B′. Also for b, b′ ∈ B′ we have F̃_{i}b = b′, if and only if Ẽ_{i}b′ = b. (iii) For each b ∈ B′, there is a unique element b̃ ∈ L′ ∩ \overline{L′} such that π′(b̃) = b. The set of elements {b̃ : b ∈ B′} forms a basis of U^{−}, the global crystal basis of U^{−}.

It was shown by Lusztig [9, 2.3] that the global crystal basis of Kashiwara coincides with the canonical basis of U^{−}.

There is a parametrization of B arising from Kashiwara’s approach, again dependent on a reduced expression i for w_{0}. Let i = (i_{1}, i_{2}, …, i_{k}) and b ∈ B. Let a_{1} be maximal such that Ẽ_{i_{1}}^{a_{1}}b ≢ 0 mod vL′; let a_{2} be maximal such that Ẽ_{i_{2}}^{a_{2}} Ẽ_{i_{1}}^{a_{1}}b ≢ 0 mod vL′, and so on, so that a_{k} is maximal such that Ẽ_{i_{k}}^{a_{k}} Ẽ_{i_{k−1}}^{a_{k−1}} ⋯ Ẽ_{i_{2}}^{a_{2}} Ẽ_{i_{1}}^{a_{1}}b ≢ 0 mod vL′. Let a = (a_{1}, a_{2}, …, a_{k}). We write ψ_{i}(b) = a. This is the crystal string of b — see [3, §2] and the end of Section 2 in [16]; see also [7]. It is known that ψ_{i}(b) uniquely determines b ∈ B (see [16, §2.5]). We have b ≡ F̃_{i_{1}}^{a_{1}} F̃_{i_{2}}^{a_{2}} ⋯ F̃_{i_{k}}^{a_{k}} · 1 mod vL′. The image of ψ_{i} is a cone which first appears in [3]. We shall call this the string cone Y_{i} = ψ_{i}(B).

We next define a function which compares Kashiwara’s approach with Lusztig’s approach. Consider the maps

Y_{i} ⟶_{ψ_{i}^{−1}} B ⟶_{ϕ_{j}} ℕ^{k}

where j = 135 ⋯ 246 ⋯ 135 ⋯ 246 · · ·. We define S_{i}^{j} = ϕ_{j}ψ_{i}^{−1} : Y_{i} → ℕ^{k}, a reparametrization function. We shall show that this function has some interesting properties in the case in which g has type A_{4}.

3 The Lusztig cones and their spanning vectors

Lusztig [12] introduced certain regions which, in low rank, give rise to canonical basis elements of a particularly simple form. We consider reduced expressions i = (i_{1}, i_{2}, …, i_{k}) for w_{0}. We shall identify this k-tuple with the reduced expression s_{i_{1}}s_{i_{2}} ⋯ s_{i_{k}}. Given two such reduced expressions we say that i ∼ i′ if there is a sequence of commutations (of the form s_{i}s_{j} = s_{j}s_{i} with |i − j| > 1) which, when applied to i, give i′. This is an equivalence relation on the set of reduced expressions for w_{0}, and the equivalence classes are called commutation classes.

The Lusztig cone, C_{i}, corresponding to a reduced expression i for w_{0} is defined to be the set of points a ∈ ℕ^{k} satisfying the following inequalities: (*) For every pair s, s′ ∈ [1, k] with s < s′, i_{s} = i_{s′} = i and i_{p} ≠ i whenever s < p < s′, we have

(∑_{p} a_{p}) − a_{s} − a_{s′} ≥ 0,

where the sum is over all p with s < p < s′ such that i_{p} is joined to i in the Dynkin diagram.

It was shown by Lusztig [12] that if a ∈ C_{i} then the monomial F_{i_{1}}^{(a_{1})}F_{i_{2}}^{(a_{2})} ⋯ F_{i_{k}}^{(a_{k})} lies in the canonical basis B, provided n = 1, 2, 3. The second author [14] showed that this remains true if n = 4. The Lusztig cones have been studied in the second author’s preprints [13] and [15] in type A for every reduced expression i for the longest word, and have also been studied by Bedard in [1] for arbitrary finite (simply-laced) type for reduced expressions compatible with a quiver whose underlying graph is the Dynkin diagram. Bedard describes these vectors using the Auslander-Reiten quiver of the quiver and homological algebra, showing they are closely connected to the representation theory of the quiver.

The reduced expression i defines an ordering on the set Φ^{+} of positive roots of the root system associated to W. We write α^{j} = s_{i_{1}}s_{i_{2}} ⋯ s_{i_{j−1}}(α_{i_{j}}) for j = 1, 2, …, k. Then Φ^{+} = {α^{1}, α^{2}, … α^{k}}. For a = (a_{1}, a_{2}, …, a_{k}) ∈ ℤ^{k}, write a_{α^{j}} = a_{j}. If α = α_{ij} := α_{i} + α_{i+1} + ⋯ + α_{j−1} with i < j, we also write a_{ij} for a_{α_{ij}}.

We can regard C_{i} as a subset of ℤ^{k} defined by the inequalities in (*) above, together with the n inequalities a_{α_{j}} ≥ 0 for j = 1, 2, …, n (see the paragraph before Lemma 4.2 in [13]). The number of inequalities in (*) is k − n, thus we have k inequalities altogether defining C_{i}. There is therefore a matrix P_{i} ∈ M_{k}(ℤ) such that

C_{i} = {a ∈ ℤ^{k} : P_{i}a ≥ 0},

where, for z ∈ ℤ^{k}, z ≥ 0 means that each entry in z is nonnegative.

There is a description of C_{i} in [13] which will be useful to us. We shall need the chamber diagram (chamber ansatz) for i defined in [2, §§1.4, 2.3]. We take n + 1 strings, numbered from top to bottom, and write i from left to right along the bottom of the diagram. Above a letter i_{j} in i, the i_{j}th and (i_{j}+1)st strings from the top above i_{j} cross. Thus, for example, in the case n = 3 with i = (1, 3, 2, 1, 3, 2), the chamber diagram is shown in Figure 3.

We denote the chamber diagram of i by CD(i). We shall be concerned with the bounded chambers of the diagram. A chamber will be defined as a pair (c, i), where c is a bounded component of the complement of CD(i). Each chamber (c, i) can be labelled with the numbers of the strings passing

1 2 3 4 1  3  2  1  3  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134  13 3 Figure 3: Chamber Diagram

below it, denoted l(c, i). Following [2], we call such a label a chamber set. For example, the chamber sets corresponding to the 3 bounded chambers in Figure 3 are 134, 3, 13.

There is another way to think of the set of bounded chambers in a chamber diagram. We recall that a quiver of type A_{n} is a directed graph such that the underlying undirected graph is the Dynkin diagram of type A_{n}. Following [13, §5.1], we define a partial quiver P of type A_{n} to be a quiver of type A_{n} which has some (or none) of its arrows replaced by undirected edges in such a way that the subgraph obtained by deleting undirected edges and vertices incident only with undirected edges is non-empty and connected. We shall now number the edges of a partial quiver of type A_{n} from 2 to n, starting from the right, as in Figure 4.

Figure 4: Edge Numbering of the Dynkin Diagram.^{∘}  ^{∘}  ^{∘}  ^{∘}  ^{∘}  ^{∘}^{2}^{3}^{n − 1}^{n}

If P is a partial quiver of type A_{n}, we denote by l(P) the subset of [1, n + 1] defined as follows: Let l_{1}(P) = {j ∈ [2, n] : edge j of P is an L} (this means that edge j has an arrow pointing to the left). If the rightmost directed edge of P is an R, and this is in position i, then let l_{2}(P) = [1, i − 1]. Otherwise l_{2}(P) is the empty set. If the leftmost directed edge of P is an R, and this is edge j, let l_{3}(P) = [j + 1, n + 1]. Otherwise l_{3}(P) is the empty set. We then define l(P) = l_{1}(P) ∪ l_{2}(P) ∪ l_{3}(P).

It is shown in [13, §5.4] that the map l is a bijection from the set of partial quivers of type A_{n} to the set of all chamber sets. For example, the chamber sets associated to the 8 partial quivers of type A_{3} are shown in Figure 5.

We now consider spanning vectors for the Lusztig cone C_{i}. The k × k matrix P_{i} has an inverse Q_{i} with entries in ℕ (see [13, §4.2]). We have k = \frac{1}{2}n(n + 1). Note that k − n = \frac{1}{2}n(n − 1) of the rows of P_{i} correspond to inequalities arising from consecutive occurrences of a letter in i. Each such consecutive pair corresponds naturally to a bounded chamber (c, i). Thus for each chamber (c, i), there is a corresponding row of P_{i} and therefore a corresponding column of Q_{i}. Now the columns of Q_{i} give spanning vectors of C_{i} in the sense that the elements of C_{i} are nonnegative linear combinations of such spanning vectors. We denote this spanning vector by a(c, i). The remaining n rows of P_{i} correspond to inequalities a_{α_{j}} ≥ 0 for α_{j} a simple root. We denote the corresponding spanning vectors by a(j, i). Thus the k spanning vectors of C_{i} are:

a(c, i)  (k − n vectors),
a(j, i)  (n vectors).

Partial Quiver  Chamber Set_{∘}^{∘}^{∘}^{∘}^{∘}^{∘}^{∘}^{∘}  _{∘}^{∘}^{∘}^{∘}^{∘}^{∘}^{∘}^{∘}  _{∘}^{∘}^{∘}^{∘}^{∘}^{∘}^{∘}^{∘}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{R}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{R}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{R}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{R}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{R}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{R}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{L}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{L}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{L}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{L}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{L}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{L}  _{{1, 3, 4}}^{{1, 2, 4}}^{{2}}^{{3}}^{{1, 4}}^{{2, 4}}^{{1, 3}}^{{2, 3}} Figure 5: Partial Quivers of Type A_{3}

It is shown in [13] how to obtain the spanning vectors a(c, i), a(j, i). We recall that the coordinates of a = (a_{1}, a_{2}, …, a_{k}) correspond naturally to the positive roots α^{1}, α^{2}, …, α^{k}, given the reduced expression i for w_{0}. It is possible to find the spanning vectors by attaching to each positive root α^{m} a multiplicity a_{m}, which will be the appropriate coordinate of the spanning vector a. It is shown in [13] that the multiplicity function α^{m} ↦ a_{m} for a(c, i) depends only upon the partial quiver P with l(P) = l(c, i), and that the multiplicity function for a(j, i) depends only upon j. These multiplicity functions are constructed by the following algorithm.

For j ∈ [1, n], the multiplicity of a positive root α_{pq} = α_{p} + α_{p+1} + ⋯ + α_{q−1} is 1 if 1 ≤ p ≤ j < j + 1 ≤ q ≤ n + 1 and is 0 otherwise. This determines the spanning vector a(j, i), whose components are all either 0 or 1.

For a partial quiver P the multiplicity of a positive root α_{pq} is given as follows. We consider the components of P, i.e. the maximal connected subquivers all of whose arrows point in the same direction. For each component Y of P let a(Y) be the number of the leftmost edge to the right of Y and let b(Y) be the number of the rightmost edge to the left of Y. We have a(Y) < b(Y). The component Y determines a set Φ^{+}(Y) of positive roots α_{pq} such that 1 ≤ p ≤ a(Y) < b(Y) ≤ q ≤ n + 1. Let m_{pq} be the number of components Y of P such that α_{pq} ∈ Φ^{+}(Y). Then the multiplicity a_{pq} of α_{pq} is given by a_{pq} = ⌈\frac{1}{2}m_{pq}⌉, which is the smallest integer m with \frac{1}{2}m_{pq} ≤ m. The multiplicity a_{pq} is the coordinate of a(c, i) corresponding to the positive root α_{pq}.

4 Transforms of the Lusztig cones and their spanning vectors

We now bring into play the parametrization of the canonical basis B arising from Kashiwara’s approach. We recall there is a bijection

ψ_{i} : B → Y_{i}

between the canonical basis and the string cone Y_{i}. Now it has been shown by the second author [15] and independently by Premat [17] that C_{i} ⊆ Y_{i}, i.e. that the Lusztig cone lies in the string cone. Thus there is a corresponding subset ψ_{i}^{−1}(C_{i}) of the canonical basis B.

We also have a bijection

ϕ_{i} : B → ℕ^{k}

and a transition function

S_{i}^{j} : Y_{i} → ℕ^{k}

given by S_{i}^{j} = ϕ_{j}ψ_{i}^{−1}. We consider the question: what is the subset S_{i}^{j}(C_{i}) of ℕ^{k} corresponding to the Lusztig cone C_{i}? We state the following conjecture:

Conjecture 4.1 For any reduced expression i of w_{0}, S_{i}^{j}(C_{i}) is a region of linearity of Lusztig’s piecewiselinear function R = R_{j}^{j′}.

We now describe an algorithm proved by the authors, as yet unpublished, which gives the transforms S_{i}^{j}(a(c, i)), S_{i}^{j}(a(j, i)) of the spanning vectors of C_{i}.

We first define a (p, q, r, s)-rectangle. Suppose we begin with the array of numbers as in Figure 6.

Figure 6: Array of numbers for defining a (p, q, r, s) rectangle. · · · · · · · · ·  · · · · · · · · · · · · · · · · · · · · · 0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2

Let (p, q, r, s) ∈ ℕ^{4}. A (p, q, r, s)-rectangle is a rectangle in this array, with top vertex on line p, middle vertices on lines q, r, and bottom vertex on line s and with all vertices lying half-way between two numbers. All lines of the rectangle are at an angle of ±π/4 to the horizontal. This forces p < q < s, p < r < s and p + s = q + r. We take only alternate columns of numbers in the rectangle, starting with the first column if q is odd, and with the second column if q is even. A (0, 3, 5, 8)-rectangle is shown in Figure 7. Let P be a partial quiver. Let Y be a component of P and let a = a(Y) and let b = b(Y) be the integers defined above. Let ρ(Y) be a (0, a, n + 2 − b, n + a − b + 2)-rectangle if Y has type L, and a (b − a − 1, b − 1, n + 1 − a, n + 1)-rectangle if Y is of type R.

Now let j ∈ [1, n]. Let a = j and b = j+1. Let ρ(j) be a (0, a, n+2−b, n+a−b+2) = (0, j, n+1−j, n+1)- rectangle (note that this is also a (b − a − 1, b − 1, n + 1 − a, n + 1)-rectangle).

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We define the diagram E(P) of P in the following way. We go through the components Y of P one by one, from left to right. It is possible to fit the rectangles ρ(Y) together as follows. If a component of type L is followed by a component of type R, the corresponding rectangles share leftmost corners, and if a component of type R is followed by a component of type L, they share rightmost corners. In each case, it is easy to see that when the rectangles are superimposed, sharing a common leftmost or rightmost corner, the overlapping numbers agree. The resulting diagram is defined to be E(P). It is convenient to label each rectangle by the corresponding component of P; on the left hand corner if Y is of type R, and on the right hand corner if Y is of type L.

We also define the diagram E(j) for each j ∈ [1, n]. E(j) is defined to be the rectangle ρ(j).

Example We consider the case when n = 6 and P = LRLL−. Then P has three components, L_{1} = L − − − −, R_{1} = − R − − − and L_{2} = − − LL−. The rectangles ρ(Y) are as in Figure 8. These rectangles fit

ρ(L_{1}) =  1 2 3 4 5^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ρ(R_{1}) =  2 3  3 4  4 5 5  6^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ρ(L_{2}) =  1 2  2 3  3 4^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Figure 8: The rectangles ρ(Y).

together to give the diagram E(P). The left hand corners of ρ(L_{1}), ρ(R_{1}) match, and so do the right hand corners of ρ(R_{1}), ρ(L_{2}). The diagram E(P) is shown in Figure 9. If P is a partial quiver, the sides

1 2 3 4 5^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  L_{1} 2 3  3 4  4 5 5  6^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} R_{1}  1 2  2 3  3 4^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  L_{2} Figure 9: The diagram E(P).

of the rectangles ρ(Y) for Y a component of P divide E(P) into diagonal rows of smaller rectangles called boxes. Consider the boxes in a diagonal running from top left to bottom right. The number of boxes in such a diagonal will be odd for the first p diagonals, starting from the top, for some p, and even for the remaining diagonals (or vice versa). Let G be the line in the diagram dividing the two adjacent diagonals containing an even and an odd number of boxes. Similarly, we can consider the boxes in a diagonal running from top right to bottom left. Again, it is possible to draw a line H separating the diagonals with an odd number of boxes from those with an even number. Let C be the point of intersection of the lines G, H. We call C the centre of the diagram E(P). Let V be the vertical line through C. In our example, the picture is shown in Figure 10. Let X be an extremal left or right

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R_{1}  1 2  2 3  3 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corner of E(P). Let R(X) be the rectangle which has the given corner as a vertex and whose edges through this point extend as far as possible in the figure (the vertex of the rectangle opposite to the given corner point may not be explicitly shown in the figure). The vertical line V divides R(X) into two parts; let Φ^{+}(X) be the set of positive roots obtained from the part of R(X) on the same side of V as X, by reading the numbers downwards in vertical lines. Thus i, i + 1, …, j gives the positive root α_{i} + α_{i+1} + ⋯ + α_{j}. Let Φ^{+}(P) be the union of Φ^{+}(X) for all left and right corners X. Thus in the example in Figure 9 we have:

Φ^{+}(L_{1}) = {α_{1} + α_{2}}
Φ^{+}(L_{2}) = {α_{2} + α_{3} + α_{4}}
Φ^{+}(R_{1}) = {α_{5}, α_{3} + α_{4} + α_{5} + α_{6}}
Φ^{+}(X_{0}) = {α_{1} + α_{2} + α_{3}}

where X_{0} is the only unlabelled left or right corner. Finally,

Φ^{+}(P) = {α_{1} + α_{2}, α_{2} + α_{3} + α_{4}, α_{5}, α_{3} + α_{4} + α_{5} + α_{6}, α_{1} + α_{2} + α_{3}}.

We also define a set Φ^{+}(j) for j ∈ [1, n]. Φ^{+}(j) is the set of positive roots obtained from the rectangle E(j) = ρ(j) by reading the numbers downwards in vertical lines.

Theorem 4.2 (Carter and Marsh). Suppose the coordinates of the vector S_{i}^{j}(a(c, i)) are labelled by the positive roots Φ^{+} by means of the reduced expression j of w_{0}. Let P be the partial quiver P (c, i). Then the coordinate of S_{i}^{j}(a(c, i)) labelled by α ∈ Φ^{+} is 1 if α ∈ Φ^{+}(P) and is 0 if α ∉ Φ^{+}(P). Also, the coordinate of S_{i}^{j}(a(j, i)) labelled by α ∈ Φ^{+} is 1 if α ∈ Φ^{+}(j) and is 0 if α ∉ Φ^{+}(j).

The proof of this result depends on Theorem 5.17 in [15], together with the description of the reparametrization function associated with Lusztig’s parametrization provided in [2]. It is hoped that this proof will appear in due course. In the present paper we shall illustrate it in type A_{4}.

5 Regions of linearity of the function R in type A_{4}

We shall now consider in detail the case when g has type A_{4}. The Dynkin diagram will be labelled as in Figure 1. In this case, j, j′ are the reduced words for w_{0} given by

j = (1, 3, 2, 4, 1, 3, 2, 4, 1, 3)
j′ = (2, 4, 1, 3, 2, 4, 1, 3, 2, 4)

The piecewise-linear function R = R_{j}^{j′} can be written as a composition of functions corresponding to a sequence of reduced words beginning with j and ending with j′ such that consecutive words differ by a braid relation. When s_{i}s_{j} is replaced by s_{j}s_{i} the corresponding pair of components a, b is replaced by b, a. When s_{i}s_{j}s_{i} is replaced by s_{j}s_{i}s_{j} the corresponding triple of components a, b, c is replaced by a′, b′, c′ where:

(a′, b′, c′) = \{ \begin{matrix}(b + c − a, a, b) & if a ≤ c \\ (b, c, b + a − c) & if a ≥ c. \end{matrix}

See [9, §2]. We list in Table I a sequence of reduced words of this kind from j to j′, underlying the long braid relations which are used. These long braid relations are denoted by the letters A, B, C, D, E, F, G, H, I, J.

Table I

Sequence of reduced words for w_{0}.

1 3 2 4 1 3 2 4 1 3 3 1 2 4 1 3 2 4 1 3

3 \underline{1 2 1} 4 3 2 4 1 3  A

3 2 1 2 4 3 2 4 1 3

3 2 1 4 \underline{2 3 2} 4 1 3  B

3 2 1 4 3 2 3 4 1 3

3 2 1 4 3 2 \underline{3 4 3} 1  C

3 2 1 4 3 2 4 3 4 1

3 2 1 \underline{4 3 4} 2 3 4 1  D

3 2 1 3 4 3 2 3 4 1

\underline{3 2 3} 1 4 3 2 3 4 1  E
2 3 2 1 4 \underline{3 2 3} 4 1  F

2 3 2 1 4 2 3 2 4 1

2 3 \underline{2 1 2} 4 3 2 4 1  G

2 3 1 2 1 4 3 2 4 1 2 3 1 2 4 1 3 2 4 1 2 3 1 2 4 3 1 2 4 1

2 3 1 2 4 3 \underline{1 2 1} 4  H

2 3 1 2 4 3 2 1 2 4

2 3 1 4 \underline{2 3 2} 1 2 4  I

2 3 1 4 3 2 3 1 2 4

2 1 \underline{3 4 3} 2 3 1 2 4  J

2 1 4 3 4 2 3 1 2 4 2 4 1 3 4 2 3 1 2 4 2 4 1 3 2 4 3 1 2 4 2 4 1 3 2 4 1 3 2 4

Let v ∈ ℝ^{10} and let R(v) be the image of v under R = R_{j}^{j′}. Since

(a′, b′, c′) = \{ \begin{matrix}(b, c, b) + (c − a, a − c, 0) & if a ≤ c \\ (b, c, b) + (0, 0, a − c) & if a ≥ c, \end{matrix}

we consider the function g(v) obtained from v by applying the map (a, b, c) ↦ (b, c, b) at each length 3 braid relation and (a, b) ↦ (b, a) at each length 2 braid relation. We write

R(v) = g(v) + ε(v).

In our case we write v = (a, b, c, d, e, f, g, h, i, j). By following the sequence of steps in Table I we see that g(v) = (c, h, e, j, g, h, i, j, g, h).

We also define vectors α_{X} for X ∈ {A, B, C, D, E, F, G, H, I, J}, by α_{X}(v) = ξ − η where ξ, η are the first and third components of the triple in the vector to which the braid relation at X is applied. These vectors α_{X} can be read off from Table II and are as follows:

α_{A}(v) = a − e
α_{B}(v) = c − g
α_{C}(v) = f − j
α_{D}(v) = d − h
α_{E}(v) = b − f
α_{F} (v) = f − j
α_{G}(v) = c − g
α_{H}(v) = e − i
α_{I}(v) = 0
α_{J}(v) = f − j

We note that the vectors α_{A}, α_{B}, α_{C}, α_{D}, α_{E}, α_{H} are linearly independent and that the remaining vectors are expressible in terms of these by

α_{J} = α_{F} = α_{C}, α_{G} = α_{B}, α_{I} = 0.

Now for each of the 10 values of X we have a choice between 2 linear functions. It may appear from this that the total number of regions of Lusztig’s function R is 2^{10}. This is not so, however, for some of the systems of inequalities determining the sequence of choices may be inconsistent. Calculation along the lines outlined in [5] shows that there are 204 consistent choices. Again it is possible for two different consistent sequences to give the same linear function R. Thus we obtain an equivalence relation on the set of consistent sequences. Calculation shows that there are 144 equivalence classes. Thus the function R in type A_{4} has 144 regions of linearity.

Each region of linearity may be defined by a system of independent inequalities of the form

∑_{X} n_{X}α_{X} ≥ 0
or  ∑_{X} n_{X}α_{X} ≤ 0

where n_{X} ∈ ℕ.

Table II

Determination of vector g(v).

a b c d e f g h i j b a c d e f g h i j

b \underline{a c e} d f g h i j  A

b c e c d f g h i j

b c e d \underline{c f g} h i j  B

b c e d f g f h i j

b c e d f g \underline{f h j} i  C

b c e d f g h j h i

b c e \underline{d f h} g j h i  D

b c e f h f g j h i

\underline{b c f} e h f g j h i  E
c f c e h \underline{f g j} h i  F

c f c e h g j g h i

c f \underline{c e g} h j g h i  G

c f e g e h j g h i c f e g h e j g h i c f e g h j e g h i

c f e g h j \underline{e g i} h  H

c f e g h j g i g h

c f e h \underline{g j g} i g h  I

c f e h j g j i g h

c e \underline{f h j} g j i g h  J

c e h j h g j i g h c h e j h g j i g h c h e j g h j i g h c h e j g h i j g h

We list the defining inequalities for the 144 regions in Table III. We use the following notation. The line

ABC, BCDEH||A, E, BCDE, BEH, AB^{2}CDEH will denote the region:

α_{A} + α_{B} + α_{C} ≤ 0  α_{A} ≥ 0
α_{B} + α_{C} + α_{D} + α_{E} + α_{H} ≤ 0  α_{E} ≥ 0
α_{B} + α_{C} + α_{D} + α_{E} ≥ 0
α_{B} + α_{E} + α_{H} ≥ 0
α_{A} + 2α_{B} + α_{C} + α_{D} + α_{E} + α_{H} ≥ 0

The values of R(v) for v in each of these 144 regions of linearity can be obtained from Table IV. We have R(v) = g(v) + ε(v) where g(v) = (c, h, e, j, g, h, i, j, g, h) and ε(v) is given in Table IV.

Table III Regions of linearity of R.

Region of linearity  Defining inequalities
1  A, BC, CD, E, BH ∥ BCD
2  A, BC, E, BH ∥ B, CD
3  ABC, CD, E, BH ∥ A, BCD
4  A, BC, E, BCDH ∥ BCD, BH
5  A, CD, E, BH ∥ BC, D
6  A, BC, CD, BEH ∥ BCD, E
7  A, BC, BEH ∥ B, CD, E
8  ABC, E, BH ∥ A, B, CD
9  ABC, E, BCDH ∥ A, BCD, BH
10  A, E, BCDH ∥ BC, D, BH
11  A, CD, BEH ∥ BC, D, E
12  A, B, C, BEH ∥ CD, BE
13  B, ABC, E, H ∥ AB, CD

Region of linearity  Defining inequalities
14  ABC, BCD, E, H ∥ ABCD, BH
15  A, D, E, BCH ∥ BC, BH
16  A, C, D, BEH ∥ BC, DE
17  A, B, BEH ∥ C, D, BE
18  AB, C, BEH ∥ A, CD, BE
19  ABC, BE, H ∥ AB, CD, E
20  B, ABC, E ∥ AB, CD, H
21  ABC, BCD, E ∥ B, ABCD, H
22  BCD, E, H ∥ ABC, D, BH
23  D, E, BCH ∥ A, BC, BH
24  A, DE, BCH ∥ BC, E, BH
25  A, C, BDEH ∥ BC, DE, BEH
26  A, D, BEH ∥ B, C, DE
27  A, B, D, BEH ∥ C, BDE
28  AB, C, BE, H ∥ CD, ABE
29  B, ABC, CD, E ∥ ABCD, H
30  BC, D, E, H ∥ ABC, BH
31  A, C, DE, BH ∥ BC, BEH
32  A, BDEH ∥ B, C, DE, BEH
33  AB, BEH ∥ A, C, D, BE
34  ABC, BE ∥ AB, CD, E, H
35  BCD, E ∥ B, ABC, D, H
36  DE, BCH ∥ A, BC, E, BH

37  C, DE, BH ∥ A, BC, BEH
38  A, DE, BH ∥ B, C, BEH
39  A, B, BDEH ∥ C, BDE, BEH
40  AB, D, BEH ∥ A, C, BDE
41  AB, BE, H ∥ C, D, ABE
42  AB, C, BE ∥ CD, ABE, H
43  ABC, CD, BE ∥ ABCD, E, H
44  B, CD, E ∥ ABC, D, H
45  BC, D, E ∥ B, ABC, H
46  BC, DE, H ∥ ABC, E, BH
47  DE, BH ∥ A, B, C, BEH
48  AB, BDEH ∥ A, C, BDE, BEH
49  AB, BE ∥ C, D, ABE, H
50  CD, BE ∥ ABC, D, E, H
51  BC, DE ∥ B, ABC, E, H
52  B, DE, H ∥ AB, C, BEH
53  AB, BDE, H ∥ C, ABDE, BEH
54  AB, D, BE ∥ C, ABDE, H
55  C, D, BE ∥ ABC, DE, H
56  B, C, DE ∥ ABC, BE, H
57  B, DE ∥ AB, C, BE, H
58  BDE, H ∥ AB, C, DE, BEH
59  AB, BDE ∥ C, BE, ABDE, H

60  D, BE ∥ AB, C, DE, H
61  C, BDE ∥ ABC, BE, DE, H
62  BDE ∥ AB, C, BE, DE, H
63  B, AB, C, BE, ABE, H ∥ CD
64  ABC, CD, BCD, E, H, BH ∥ ABCD
65  BC, ABC, BCD, ABCD, E, H ∥ BH
66  A, C, D, DE, BH, BEH ∥ BC
67  A, C, BC, CD, BCD, BEH ∥ BCDE
68  BC, ABC, BCDE, ABCDE, H ∥ E, BH
69  B, C, ABC, BCDE, ABCDE ∥ BE, H
70  BC, ABC, BCD, ABCD, E ∥ B, H
71  B, AB, C, BE, ABE ∥ CD, H
72  B, AB, BDE, ABDE, H ∥ C, BEH
73  B, AB, BE, ABE, H ∥ C, D
74  A, D, DE, BH, BEH ∥ B, C
75  C, D, DE, BH, BEH ∥ A, BC
76  B, D, DE, H, BEH ∥ AB, C
77  C, BC, DE, H, BH ∥ ABC, BEH
78  CD, BCD, E, H, BH ∥ ABC, D
79  B, AB, D, BE, ABDE ∥ C, H
80  B, C, D, BE, DE ∥ ABC, H
81  C, ABC, CD, ABCD, BE ∥ ABCDE, H
82  C, D, BCDE, H, BEH ∥ ABC, DE

83  AB, D, BDE, H, BEH ∥ C, ABDE
84  A, C, BC, BCDEH, B^{2}CDEH ∥ BCDE, BEH
85  C, ABC, CD, ABCD, BEH ∥ A, BCDE
86  C, ABC, BCDE, H, AB^{2}CDEH ∥ ABCDE, BEH
87  ABC, CD, BCDE, H, BEH ∥ ABCD, E
88  BC, ABC, BCDE, ABCDE ∥ B, E, H
89  D, DE, BH, BEH ∥ A, B, C
90  B, AB, BDE, ABDE ∥ C, BE, H
91  B, AB, BE, ABE ∥ C, D, H
92  B, D, DE, BE ∥ AB, C, H
93  C, ABC, BCDE, AB^{2}CDE ∥ BE, ABCDE, H
94  C, BCDE, H, BDEH ∥ ABC, DE, BEH
95  D, BDE, H, BEH ∥ AB, C, DE
96  CD, BCDE, H, BEH ∥ ABC, D, E
97  C, ABC, BCDEH, AB^{2}CDEH ∥ A, BCDE, BEH
98  A, D, E ∥ B, BC, BH, BCH
99  B, E, H ∥ AB, ABC, D, CD
100  CD, E, BH ∥ A, ABC, D, BCD
101  A, E, BH ∥ B, BC, D, CD
102  A, BC, E ∥ B, BCD, BH, BCDH
103  ABC, CD, BEH ∥ A, ABCD, E, BCDE
104  ABC, BCDE, H ∥ E, ABCDE, BEH, AB^{2}CDEH
105  A, BC, BCDEH ∥ E, BCDE, BEH, B^{2}CDEH

106  C, D, BEH ∥ A, ABC, DE, BCDE
107  A, B, C ∥ BE, BCDE, BEH, BCDEH
108  B, E ∥ AB, ABC, D, CD, H
109  A, E ∥ B, BC, D, BH, BCDH
110  D, E ∥ A, B, BC, BH, BCH
111  A, DE ∥ B, BC, E, BH, BCH
112  E, BCDH ∥ A, ABC, D, BCD, BH
113  E, BH ∥ A, B, ABC, D, CD
114  ABC, E ∥ A, B, BCD, BH, BCDH
115  ABC, BEH ∥ A, AB, CD, E, BE
116  ABC, BCDEH ∥ A, E, BCDE, BEH, AB^{2}CDEH
117  BE, H ∥ AB, ABC, D, CD, E
118  A, BEH ∥ B, BC, D, CD, E
119  CD, BEH ∥ A, ABC, D, E, BCDE
120  D, BEH ∥ A, AB, C, DE, BDE
121  C, BDEH ∥ A, ABC, DE, BCDE, BEH
122  BCDE, H ∥ ABC, E, DE, BEH, BDEH
123  A, BCDEH ∥ BC, E, DE, BEH, BDEH
124  ABC, BCDE ∥ ABCDE, AB^{2}CDE, E, BE, H
125  AB, C ∥ A, BE, BCDE, BEH, BCDEH
126  A, BC ∥ B, E, BCDE, BEH, BCDEH
127  A, B ∥ C, BE, BDE, BEH, BDEH
128  DE ∥ A, B, BC, E, BH, BCH

129  BDEH ∥ A, AB, C, DE, BDE, BEH
130  AB ∥ A, C, BE, BDE, BEH, BDEH
131  BE ∥ AB, ABC, D, CD, E, H
132  BCDE ∥ ABC, E, BE, DE, BDE, H
133  C, BC, ABC, BCDE, ABCDE, H, BH ∥ BEH
134  B, C, ABC, CD, ABCD, BE, ABCDE ∥ H
135  B, AB, D, BDE, ABDE, H, BEH ∥ C
136  C, BC, D, DE, H, BH, BEH ∥ ABC
137  C, ABC, CD, ABCD, BCDE, H, BEH ∥ ABCDE
138  E ∥ A, B, ABC, D, BCD, BH, BCDH
139  BEH ∥ A, AB, ABC, D, CD, E, BE
140  BCDEH ∥ A, ABC, E, DE, BCDE, BEH, BDEH
141  ABC ∥ A, AB, E, BE, BCDE, BEH, BCDEH
142  A ∥ B, BC, E, DE, BEH, BDEH, BCDEH
143  C, BC, ABC, CD, BCD, ABCD, BCDE, ABCDE, H, BH, BEH ∥ -
144  - ∥ A, AB, ABC, E, BE, DE, BDE, BCDE, BEH, BDEH, BCDEH.

Table IV The linear function ε(v) on each region.

Region of linearity  ε(v)
1  (−α_{A} − α_{E}, −α_{B} + α_{D} − α_{E} − α_{H}, α_{A}, α_{C} + α_{E}, −α_{C} − α_{D}, −α_{B} − α_{C},

α_{B} + α_{C} + α_{D} + α_{H}, α_{B} + 2α_{C} + α_{D}, 0, 0),

2  (−α_{A} − α_{E}, −α_{B} + α_{D} − α_{E} − α_{H}, α_{A}, α_{C} + α_{E}, 0, −α_{B} − α_{C}, α_{B} + α_{H},

α_{B} + α_{C}, α_{C} + α_{D}, 0),

Region of linearity  ε(v)
3  (−α_{E}, −α_{B} + α_{D} − α_{E} − α_{H}, 0, α_{C} + α_{E}, −α_{C} − α_{D}, −α_{A} − α_{B} − α_{C},

α_{B} + α_{C} + α_{D} + α_{H}, α_{A} + α_{B} + 2α_{C} + α_{D}, 0, 0),

4  (−α_{A} − α_{E}, α_{D} − α_{E}, α_{A}, α_{C} + α_{E}, −α_{B} − α_{C} − α_{D} − α_{H}, −α_{B} − α_{C},

α_{B} + α_{C} + α_{D} + α_{H}, 2α_{B} + 2α_{C} + α_{D} + α_{H}, 0, 0),

5  (−α_{A} − α_{E}, −α_{B} + α_{D} − α_{E} − α_{H}, α_{A}, α_{C} + α_{E}, −α_{C} − α_{D}, 0,

α_{B} + α_{C} + α_{D} + α_{H}, α_{C} + α_{D}, 0, α_{B} + α_{C}),

6  (−α_{A}, −α_{B} + α_{D} − α_{E} − α_{H}, α_{A}, α_{C}, −α_{C} − α_{D}, −α_{B} − α_{C},

α_{B} + α_{C} + α_{D} + α_{E} + α_{H}, α_{B} + 2α_{C} + α_{D}, 0, 0),

7  (−α_{A}, −α_{B} + α_{D} − α_{E} − α_{H}, α_{A}, α_{C}, 0, −α_{B} − α_{C}, α_{B} + α_{E} + α_{H}, α_{B} + α_{C},

α_{C} + α_{D}, 0),

8  (−α_{E}, −α_{B} + α_{D} − α_{E} − α_{H}, 0, α_{C} + α_{E}, 0, −α_{A} − α_{B} − α_{C}, α_{B} + α_{H},

α_{A} + α_{B} + α_{C}, α_{C} + α_{D}, 0),

9  (−α_{E}, α_{D} − α_{E}, 0, α_{C} + α_{E}, −α_{B} − α_{C} − α_{D} − α_{H}, −α_{A} − α_{B} − α_{C},

α_{B} + α_{C} + α_{D} + α_{H}, α_{A} + 2α_{B} + 2α_{C} + α_{D} + α_{H}, 0, 0),

10  (−α_{A} − α_{E}, α_{D} − α_{E}, α_{A}, α_{C} + α_{E}, −α_{B} − α_{C} − α_{D} − α_{H}, 0,

α_{B} + α_{C} + α_{D} + α_{H}, α_{B} + α_{C} + α_{D} + α_{H}, 0, α_{B} + α_{C}),

11  (−α_{A}, −α_{B} + α_{D} − α_{E} − α_{H}, α_{A}, α_{C}, −α_{C} − α_{D}, 0,

α_{B} + α_{C} + α_{D} + α_{E} + α_{H}, α_{C} + α_{D}, 0, α_{B} + α_{C}),

12  (−α_{A}, −α_{B} + α_{D} − α_{E} − α_{H}, α_{A}, −α_{B} + α_{C}, α_{B}, −α_{C}, α_{B} + α_{E} + α_{H}, α_{C},

α_{C} + α_{D}, 0),

13  (−α_{E}, −α_{B} + α_{D} − α_{E} − α_{H}, −α_{B}, α_{C} + α_{E}, α_{B}, −α_{A} − α_{B} − α_{C}, α_{H},

α_{A} + α_{B} + α_{C}, α_{C} + α_{D}, 0),

14  (−α_{E}, α_{D} − α_{E}, −α_{B} − α_{C} − α_{D}, α_{C} + α_{E}, −α_{H}, −α_{A} − α_{B} − α_{C}, α_{H},

α_{A} + 2α_{B} + 2α_{C} + α_{D} + α_{H}, 0, 0),

15  (−α_{A} − α_{D} − α_{E}, α_{D} − α_{E}, α_{A}, α_{C} + α_{E}, −α_{B} − α_{C} − α_{H}, α_{D}, α_{B} + α_{C} + α_{H},

α_{B} + α_{C} + α_{H}, 0, α_{B} + α_{C}),

16  (−α_{A}, −α_{B} + α_{D} − α_{E} − α_{H}, α_{A}, α_{C} − α_{D}, −α_{C}, α_{D},

α_{B} + α_{C} + α_{D} + α_{E} + α_{H}, α_{C}, 0, α_{B} + α_{C}),

17  (−α_{A}, −α_{B} + α_{D} − α_{E} − α_{H}, α_{A}, −α_{B} + α_{C}, α_{B}, 0, α_{B} + α_{E} + α_{H}, 0,

α_{C} + α_{D}, α_{C}),

18  (0, −α_{B} + α_{D} − α_{E} − α_{H}, 0, −α_{A} − α_{B} + α_{C}, α_{A} + α_{B}, −α_{C}, α_{B} + α_{E} + α_{H},

α_{C}, α_{C} + α_{D}, 0),

19  (0, −α_{B} + α_{D} − α_{E} − α_{H}, −α_{B} − α_{E}, α_{C}, α_{B} + α_{E}, −α_{A} − α_{B} − α_{C}, α_{H},

α_{A} + α_{B} + α_{C}, α_{C} + α_{D}, 0),

20  (−α_{E}, −α_{B} + α_{D} − α_{E}, −α_{B}, α_{C} + α_{E}, α_{B}, −α_{A} − α_{B} − α_{C}, 0, α_{A} + α_{B} + α_{C},

α_{C} + α_{D} + α_{H}, 0),

21  (−α_{E}, α_{D} − α_{E}, −α_{B} − α_{C} − α_{D}, α_{C} + α_{E}, 0, −α_{A} − α_{B} − α_{C}, 0,

α_{A} + 2α_{B} + 2α_{C} + α_{D}, α_{H}, 0),

22  (−α_{E}, α_{D} − α_{E}, −α_{B} − α_{C} − α_{D}, α_{C} + α_{E}, −α_{H}, 0, α_{H}, α_{B} + α_{C} + α_{D} + α_{H},

0, α_{A} + α_{B} + α_{C}),

23  (−α_{D} − α_{E}, α_{D} − α_{E}, 0, α_{C} + α_{E}, −α_{B} − α_{C} − α_{H}, α_{D}, α_{B} + α_{C} + α_{H},

α_{B} + α_{C} + α_{H}, 0, α_{A} + α_{B} + α_{C}),

24  (−α_{A} − α_{D} − α_{E}, α_{D}, α_{A}, α_{C}, −α_{B} − α_{C} − α_{H}, α_{D} + α_{E}, α_{B} + α_{C} + α_{H},

α_{B} + α_{C} + α_{H}, 0, α_{B} + α_{C}),

25  (−α_{A}, α_{D}, α_{A}, −α_{B} + α_{C} − α_{D} − α_{E} − α_{H}, −α_{C}, α_{B} + α_{D} + α_{E} + α_{H},

α_{B} + α_{C} + α_{D} + α_{E} + α_{H}, α_{C}, 0, α_{B} + α_{C}),

26  (−α_{A}, −α_{B} + α_{D} − α_{E} − α_{H}, α_{A}, α_{C} − α_{D}, 0, α_{D}, α_{B} + α_{D} + α_{E} + α_{H}, 0,
α_{C}, α_{B} + α_{C}),
27  (−α_{A}, −α_{B} + α_{D} − α_{E} − α_{H}, α_{A}, −α_{B} + α_{C} − α_{D}, α_{B}, α_{D},

α_{B} + α_{D} + α_{E} + α_{H}, 0, α_{C}, α_{C}),

28  (0, −α_{B} + α_{D} − α_{E} − α_{H}, −α_{B} − α_{E}, −α_{A} − α_{B} + α_{C}, α_{A} + 2α_{B} + α_{E}, −α_{C},

α_{H}, α_{C}, α_{C} + α_{D}, 0),

29  (−α_{E}, −α_{B} + α_{D} − α_{E}, −α_{B} − α_{C} − α_{D}, α_{C} + α_{E}, α_{B}, −α_{A} − α_{B} − α_{C}, 0,

α_{A} + α_{B} + 2α_{C} + α_{D}, α_{H}, 0),

30  (−α_{D} − α_{E}, α_{D} − α_{E}, −α_{B} − α_{C}, α_{C} + α_{E}, −α_{H}, α_{D}, α_{H}, α_{B} + α_{C} + α_{H}, 0,
α_{A} + α_{B} + α_{C}),
31  (−α_{A} − α_{D} − α_{E}, α_{D}, α_{A}, −α_{B} + α_{C} − α_{H}, −α_{C}, α_{B} + α_{D} + α_{E} + α_{H},

α_{B} + α_{C} + α_{H}, α_{C}, 0, α_{B} + α_{C}),

32  (−α_{A}, α_{D}, α_{A}, −α_{B} + α_{C} − α_{D} − α_{E} − α_{H}, 0, α_{B} + α_{D} + α_{E} + α_{H},

α_{B} + α_{D} + α_{E} + α_{H}, 0, α_{C}, α_{B} + α_{C}),

33  (0, −α_{B} + α_{D} − α_{E} − α_{H}, 0, −α_{A} − α_{B} + α_{C}, α_{A} + α_{B}, 0, α_{B} + α_{E} + α_{H},
0, α_{C} + α_{D}, α_{C}),
34  (0, −α_{B} + α_{D} − α_{E}, −α_{B} − α_{E}, α_{C}, α_{B} + α_{E}, −α_{A} − α_{B} − α_{C}, 0,

α_{A} + α_{B} + α_{C}, α_{C} + α_{D} + α_{H}, 0),

35  (−α_{E}, α_{D} − α_{E}, −α_{B} − α_{C} − α_{D}, α_{C} + α_{E}, 0, 0, 0, α_{B} + α_{C} + α_{D}, α_{H},
α_{A} + α_{B} + α_{C}),
36  (−α_{D} − α_{E}, α_{D}, 0, α_{C}, −α_{B} − α_{C} − α_{H}, α_{D} + α_{E}, α_{B} + α_{C} + α_{H},

α_{B} + α_{C} + α_{H}, 0, α_{A} + α_{B} + α_{C}),

37  (−α_{D} − α_{E}, α_{D}, 0, −α_{B} + α_{C} − α_{H}, −α_{C}, α_{B} + α_{D} + α_{E} + α_{H},

α_{B} + α_{C} + α_{H}, α_{C}, 0, α_{A} + α_{B} + α_{C}),

38  (−α_{A} − α_{D} − α_{E}, α_{D}, α_{A}, −α_{B} + α_{C} − α_{H}, 0, α_{B} + α_{D} + α_{E} + α_{H}, α_{B} + α_{H},

0, α_{C}, α_{B} + α_{C}),

39  (−α_{A}, α_{D}, α_{A}, −2α_{B} + α_{C} − α_{D} − α_{E} − α_{H}, α_{B}, α_{B} + α_{D} + α_{E} + α_{H},

α_{B} + α_{D} + α_{E} + α_{H}, 0, α_{C}, α_{C}),

40  (0, −α_{B} + α_{D} − α_{E} − α_{H}, 0, −α_{A} − α_{B} + α_{C} − α_{D}, α_{A} + α_{B}, α_{D},

α_{B} + α_{D} + α_{E} + α_{H}, 0, α_{C}, α_{C}),

41  (0, −α_{B} + α_{D} − α_{E} − α_{H}, −α_{B} − α_{E}, −α_{A} − α_{B} + α_{C}, α_{A} + 2α_{B} + α_{E}, 0,

α_{H}, 0, α_{C} + α_{D}, α_{C}),

42  (0, −α_{B} + α_{D} − α_{E}, −α_{B} − α_{E}, −α_{A} − α_{B} + α_{C}, α_{A} + 2α_{B} + α_{E}, −α_{C}, 0,

α_{C}, α_{C} + α_{D} + α_{H}, 0),

43  (0, −α_{B} + α_{D} − α_{E}, −α_{B} − α_{C} − α_{D} − α_{E}, α_{C}, α_{B} + α_{E}, −α_{A} − α_{B} − α_{C}, 0,

α_{A} + α_{B} + 2α_{C} + α_{D}, α_{H}, 0),

44  (−α_{E}, −α_{B} + α_{D} − α_{E}, −α_{B} − α_{C} − α_{D}, α_{C} + α_{E}, α_{B}, 0, 0, α_{C} + α_{D}, α_{H},

α_{A} + α_{B} + α_{C}),

45  (−α_{D} − α_{E}, α_{D} − α_{E}, −α_{B} − α_{C}, α_{C} + α_{E}, 0, α_{D}, 0, α_{B} + α_{C}, α_{H},

α_{A} + α_{B} + α_{C}),

46  (−α_{D} − α_{E}, α_{D}, −α_{B} − α_{C}, α_{C}, −α_{H}, α_{D} + α_{E}, α_{H}, α_{B} + α_{C} + α_{H}, 0,

α_{A} + α_{B} + α_{C}),

47  (−α_{D} − α_{E}, α_{D}, 0, −α_{B} + α_{C} − α_{H}, 0, α_{B} + α_{D} + α_{E} + α_{H}, α_{B} + α_{H}, 0,

α_{C}, α_{A} + α_{B} + α_{C}),

48  (0, α_{D}, 0, −α_{A} − 2α_{B} + α_{C} − α_{D} − α_{E} − α_{H}, α_{A} + α_{B},

α_{B} + α_{D} + α_{E} + α_{H}, α_{B} + α_{D} + α_{E} + α_{H}, 0, α_{C}, α_{C}),

49  (0, −α_{B} + α_{D} − α_{E}, −α_{B} − α_{E}, −α_{A} − α_{B} + α_{C}, α_{A} + 2α_{B} + α_{E}, 0, 0, 0,
α_{C} + α_{D} + α_{H}, α_{C}),
50  (0, −α_{B} + α_{D} − α_{E}, −α_{B} − α_{C} − α_{D} − α_{E}, α_{C}, α_{B} + α_{E}, 0, 0, α_{C} + α_{D},
α_{H}, α_{A} + α_{B} + α_{C}),
51  (−α_{D} − α_{E}, α_{D}, −α_{B} − α_{C}, α_{C}, 0, α_{D} + α_{E}, 0, α_{B} + α_{C}, α_{H},
α_{A} + α_{B} + α_{C}),
52  (−α_{D} − α_{E}, α_{D}, −α_{B}, −α_{B} + α_{C} − α_{H}, α_{B}, α_{B} + α_{D} + α_{E} + α_{H}, α_{H}, 0, α_{C},
α_{A} + α_{B} + α_{C}),
53  (0, α_{D}, −α_{B} − α_{D} − α_{E}, −α_{A} − 2α_{B} + α_{C} − α_{D} − α_{E} − α_{H},

α_{A} + 2α_{B} + α_{D} + α_{E}, α_{B} + α_{D} + α_{E} + α_{H}, α_{H}, 0, α_{C}, α_{C}),

54  (0, −α_{B} + α_{D} − α_{E}, −α_{B} − α_{D} − α_{E}, −α_{A} − α_{B} + α_{C} − α_{D},

α_{A} + 2α_{B} + α_{D} + α_{E}, α_{D}, 0, 0, α_{C} + α_{H}, α_{C}),

55  (0, −α_{B} + α_{D} − α_{E}, −α_{B} − α_{C} − α_{D} − α_{E}, α_{C} − α_{D}, α_{B} + α_{D} + α_{E}, α_{D},

0, α_{C}, α_{H}, α_{A} + α_{B} + α_{C}),

56  (−α_{D} − α_{E}, α_{D}, −α_{B} − α_{C}, −α_{B} + α_{C}, α_{B}, α_{B} + α_{D} + α_{E}, 0, α_{C}, α_{H},
α_{A} + α_{B} + α_{C}),
57  (−α_{D} − α_{E}, α_{D}, −α_{B}, −α_{B} + α_{C}, α_{B}, α_{B} + α_{D} + α_{E}, 0, 0, α_{C} + α_{H},
α_{A} + α_{B} + α_{C}),
58  (0, α_{D}, −α_{B} − α_{D} − α_{E}, −α_{B} + α_{C} − α_{D} − α_{E} − α_{H}, α_{B} + α_{D} + α_{E},

α_{B} + α_{D} + α_{E} + α_{H}, α_{H}, 0, α_{C}, α_{A} + α_{B} + α_{C}),

59  (0, α_{D}, −α_{B} − α_{D} − α_{E}, −α_{A} − 2α_{B} + α_{C} − α_{D} − α_{E},

α_{A} + 2α_{B} + α_{D} + α_{E}, α_{B} + α_{D} + α_{E}, 0, 0, α_{C} + α_{H}, α_{C}),

60  (0, −α_{B} + α_{D} − α_{E}, −α_{B} − α_{D} − α_{E}, α_{C} − α_{D}, α_{B} + α_{D} + α_{E}, α_{D}, 0,

0, α_{C} + α_{H}, α_{A} + α_{B} + α_{C}),

61  (0, α_{D}, −α_{B} − α_{C} − α_{D} − α_{E}, −α_{B} + α_{C} − α_{D} − α_{E}, α_{B} + α_{D} + α_{E},

α_{B} + α_{D} + α_{E}, 0, α_{C}, α_{H}, α_{A} + α_{B} + α_{C}),

62  (0, α_{D}, −α_{B} − α_{D} − α_{E}, −α_{B} + α_{C} − α_{D} − α_{E}, α_{B} + α_{D} + α_{E},

α_{B} + α_{D} + α_{E}, 0, 0, α_{C} + α_{H}, α_{A} + α_{B} + α_{C}),

63  (−α_{A} − α_{B} − α_{E}, −α_{B} + α_{D} − α_{E} − α_{H}, α_{A}, α_{C} + α_{E}, α_{B}, −α_{C}, α_{H}, α_{C},

α_{C} + α_{D}, 0),

64  (−α_{E}, −α_{B} + α_{D} − α_{E} − α_{H}, −α_{B} − α_{C} − α_{D}, α_{C} + α_{E}, α_{B}, −α_{A} − α_{B} − α_{C},

α_{H}, α_{A} + α_{B} + 2α_{C} + α_{D}, 0, 0),

65  (−α_{A} − α_{B} − α_{C} − α_{D} − α_{E}, α_{D} − α_{E}, α_{A}, α_{C} + α_{E}, −α_{H}, α_{D}, α_{H},

α_{B} + α_{C} + α_{H}, 0, 0),

66  (−α_{A} − α_{D} − α_{E}, −α_{B} + α_{D} − α_{E} − α_{H}, α_{A}, α_{C} + α_{E}, −α_{C}, α_{D},

α_{B} + α_{C} + α_{H}, α_{C}, 0, α_{B} + α_{C}),

67  (−α_{A}, −α_{B} + α_{D} − α_{E} − α_{H}, α_{A}, −α_{B} − α_{D}, α_{B}, α_{D},

α_{B} + α_{C} + α_{D} + α_{E} + α_{H}, α_{C}, 0, 0),

68  (−α_{A} − α_{B} − α_{C} − α_{D} − α_{E}, α_{D}, α_{A}, α_{C}, −α_{H}, α_{D} + α_{E}, α_{H}, α_{B} + α_{C} + α_{H},

0, 0),

69  (−α_{A} − α_{B} − α_{C} − α_{D} − α_{E}, α_{D}, α_{A}, −α_{B} + α_{C}, α_{B}, α_{B} + α_{D} + α_{E}, 0, α_{C},

α_{H}, 0),

70  (−α_{A} − α_{B} − α_{C} − α_{D} − α_{E}, α_{D} − α_{E}, α_{A}, α_{C} + α_{E}, 0, α_{D}, 0, α_{B} + α_{C},

α_{H}, 0),

71  (−α_{A} − α_{B} − α_{E}, −α_{B} + α_{D} − α_{E}, α_{A}, α_{C} + α_{E}, α_{B}, −α_{C}, 0, α_{C},

α_{C} + α_{D} + α_{H}, 0),

72  (−α_{A} − α_{B} − α_{D} − α_{E}, α_{D}, α_{A}, −α_{B} + α_{C} − α_{H}, α_{B}, α_{B} + α_{D} + α_{E} + α_{H}, α_{H},
0, α_{C}, α_{C}),
73  (−α_{A} − α_{B} − α_{E}, −α_{B} + α_{D} − α_{E} − α_{H}, α_{A}, α_{C} + α_{E}, α_{B}, 0, α_{H}, 0,
α_{C} + α_{D}, α_{C}),
74  (−α_{A} − α_{D} − α_{E}, −α_{B} + α_{D} − α_{E} − α_{H}, α_{A}, α_{C} + α_{E}, 0, α_{D}, α_{B} + α_{H}, 0,
α_{C}, α_{B} + α_{C}),
75  (−α_{D} − α_{E}, −α_{B} + α_{D} − α_{E} − α_{H}, 0, α_{C} + α_{E}, −α_{C}, α_{D}, α_{B} + α_{C} + α_{H}, α_{C},
0, α_{A} + α_{B} + α_{C}),
76  (−α_{D} − α_{E}, −α_{B} + α_{D} − α_{E} − α_{H}, −α_{B}, α_{C} + α_{E}, α_{B}, α_{D}, α_{H}, 0, α_{C},
α_{A} + α_{B} + α_{C}),
77  (−α_{D} − α_{E}, α_{D}, −α_{B} − α_{C}, −α_{B} + α_{C} − α_{H}, α_{B}, α_{B} + α_{D} + α_{E} + α_{H}, α_{H}, α_{C},
0, α_{A} + α_{B} + α_{C}),
78  (−α_{E}, −α_{B} + α_{D} − α_{E} − α_{H}, −α_{B} − α_{C} − α_{D}, α_{C} + α_{E}, α_{B}, 0, α_{H}, α_{C} + α_{D},
0, α_{A} + α_{B} + α_{C}),
79  (−α_{A} − α_{B} − α_{D} − α_{E}, −α_{B} + α_{D} − α_{E}, α_{A}, α_{C} + α_{E}, α_{B}, α_{D}, 0, 0,
α_{C} + α_{H}, α_{C}),
80  (−α_{D} − α_{E}, −α_{B} + α_{D} − α_{E}, −α_{B} − α_{C}, α_{C} + α_{E}, α_{B}, α_{D}, 0, α_{C}, α_{H},
α_{A} + α_{B} + α_{C}),
81  (0, −α_{B} + α_{D} − α_{E}, −α_{B} − α_{C} − α_{D} − α_{E}, −α_{A} − α_{B} − α_{D},

α_{A} + 2α_{B} + α_{C} + α_{D} + α_{E}, α_{D}, 0, α_{C}, α_{H}, 0),

82  (0, −α_{B} + α_{D} − α_{E} − α_{H}, −α_{B} − α_{C} − α_{D} − α_{E}, α_{C} − α_{D}, α_{B} + α_{D} + α_{E},

α_{D}, α_{H}, α_{C}, 0, α_{A} + α_{B} + α_{C}),

83  (0, −α_{B} + α_{D} − α_{E} − α_{H}, −α_{B} − α_{D} − α_{E}, −α_{A} − α_{B} + α_{C} − α_{D},

α_{A} + 2α_{B} + α_{D} + α_{E}, α_{D}, α_{H}, 0, α_{C}, α_{C}),

84  (−α_{A}, α_{D}, α_{A}, −2α_{B} − α_{D} − α_{E} − α_{H}, α_{B}, α_{B} + α_{D} + α_{E} + α_{H},

α_{B} + α_{C} + α_{D} + α_{E} + α_{H}, α_{C}, 0, 0),

85  (0, −α_{B} + α_{D} − α_{E} − α_{H}, 0, −α_{A} − α_{B} − α_{D}, α_{A} + α_{B}, α_{D},

α_{B} + α_{C} + α_{D} + α_{E} + α_{H}, α_{C}, 0, 0),

86  (0, α_{D}, −α_{B} − α_{C} − α_{D} − α_{E}, −α_{A} − 2α_{B} − α_{D} − α_{E} − α_{H},

α_{A} + 2α_{B} + α_{C} + α_{D} + α_{E}, α_{B} + α_{D} + α_{E} + α_{H}, α_{H}, α_{C}, 0, 0),

87  (0, −α_{B} + α_{D} − α_{E} − α_{H}, −α_{B} − α_{C} − α_{D} − α_{E}, α_{C}, α_{B} + α_{E},

−α_{A} − α_{B} − α_{C}, α_{H}, α_{A} + α_{B} + 2α_{C} + α_{D}, 0, 0),

88  (−α_{A} − α_{B} − α_{C} − α_{D} − α_{E}, α_{D}, α_{A}, α_{C}, 0, α_{D} + α_{E}, 0, α_{B} + α_{C}, α_{H}, 0),
89  (−α_{D} − α_{E}, −α_{B} + α_{D} − α_{E} − α_{H}, 0, α_{C} + α_{E}, 0, α_{D}, α_{B} + α_{H}, 0, α_{C},
α_{A} + α_{B} + α_{C}),
90  (−α_{A} − α_{B} − α_{D} − α_{E}, α_{D}, α_{A}, −α_{B} + α_{C}, α_{B}, α_{B} + α_{D} + α_{E}, 0, 0,
α_{C} + α_{H}, α_{C}),
91  (−α_{A} − α_{B} − α_{E}, −α_{B} + α_{D} − α_{E}, α_{A}, α_{C} + α_{E}, α_{B}, 0, 0, 0,
α_{C} + α_{D} + α_{H}, α_{C}),
92  (−α_{D} − α_{E}, −α_{B} + α_{D} − α_{E}, −α_{B}, α_{C} + α_{E}, α_{B}, α_{D}, 0, 0, α_{C} + α_{H},
α_{A} + α_{B} + α_{C}),
93  (0, α_{D}, −α_{B} − α_{C} − α_{D} − α_{E}, −α_{A} − 2α_{B} − α_{D} − α_{E},

α_{A} + 2α_{B} + α_{C} + α_{D} + α_{E}, α_{B} + α_{D} + α_{E}, 0, α_{C}, α_{H}, 0),

94  (0, α_{D}, −α_{B} − α_{C} − α_{D} − α_{E}, −α_{B} + α_{C} − α_{D} − α_{E} − α_{H}, α_{B} + α_{D} + α_{E},

α_{B} + α_{D} + α_{E} + α_{H}, α_{H}, α_{C}, 0, α_{A} + α_{B} + α_{C}),

95  (0, −α_{B} + α_{D} − α_{E} − α_{H}, −α_{B} − α_{D} − α_{E}, α_{C} − α_{D}, α_{B} + α_{D} + α_{E}, α_{D},

α_{H}, 0, α_{C}, α_{A} + α_{B} + α_{C}),

96  (0, −α_{B} + α_{D} − α_{E} − α_{H}, −α_{B} − α_{C} − α_{D} − α_{E}, α_{C}, α_{B} + α_{E}, 0, α_{H},

α_{C} + α_{D}, 0, α_{A} + α_{B} + α_{C}),

97  (0, α_{D}, 0, −α_{A} − 2α_{B} − α_{D} − α_{E} − α_{H}, α_{A} + α_{B}, α_{B} + α_{D} + α_{E} + α_{H},

α_{B} + α_{C} + α_{D} + α_{E} + α_{H}, α_{C}, 0, 0),

98  (−α_{A} − α_{D} − α_{E}, α_{D} − α_{E}, α_{A}, α_{C} + α_{E}, 0, α_{D}, 0, 0, α_{B} + α_{C} + α_{H},

α_{B} + α_{C}),

99  (−α_{E}, −α_{B} + α_{D} − α_{E} − α_{H}, −α_{B}, α_{C} + α_{E}, α_{B}, 0, α_{H}, 0, α_{C} + α_{D},

α_{A} + α_{B} + α_{C}),

100  (−α_{E}, −α_{B} + α_{D} − α_{E} − α_{H}, 0, α_{C} + α_{E}, −α_{C} − α_{D}, 0,

α_{B} + α_{C} + α_{D} + α_{H}, α_{C} + α_{D}, 0, α_{A} + α_{B} + α_{C}),

101  (−α_{A} − α_{E}, −α_{B} + α_{D} − α_{E} − α_{H}, α_{A}, α_{C} + α_{E}, 0, 0, α_{B} + α_{H}, 0,

α_{C} + α_{D}, α_{B} + α_{C}),

102  (−α_{A} − α_{E}, α_{D} − α_{E}, α_{A}, α_{C} + α_{E}, 0, −α_{B} − α_{C}, 0, α_{B} + α_{C},

α_{B} + α_{C} + α_{D} + α_{H}, 0),

103  (0, −α_{B} + α_{D} − α_{E} − α_{H}, 0, α_{C}, −α_{C} − α_{D}, −α_{A} − α_{B} − α_{C},

α_{B} + α_{C} + α_{D} + α_{E} + α_{H}, α_{A} + α_{B} + 2α_{C} + α_{D}, 0, 0),

104  (0, α_{D}, −α_{B} − α_{C} − α_{D} − α_{E}, α_{C}, −α_{H}, −α_{A} − α_{B} − α_{C}, α_{H},

α_{A} + 2α_{B} + 2α_{C} + α_{D} + α_{E} + α_{H}, 0, 0),

105  (−α_{A}, α_{D}, α_{A}, α_{C}, −α_{B} − α_{C} − α_{D} − α_{E} − α_{H}, −α_{B} − α_{C},

α_{B} + α_{C} + α_{D} + α_{E} + α_{H}, 2α_{B} + 2α_{C} + α_{D} + α_{E} + α_{H}, 0, 0),

106  (0, −α_{B} + α_{D} − α_{E} − α_{H}, 0, α_{C} − α_{D}, −α_{C}, α_{D},

α_{B} + α_{C} + α_{D} + α_{E} + α_{H}, α_{C}, 0, α_{A} + α_{B} + α_{C}),

107  (−α_{A}, α_{D}, α_{A}, −α_{B} + α_{C}, α_{B}, −α_{C}, 0, α_{C}, α_{B} + α_{C} + α_{D} + α_{E} + α_{H}, 0),
108  (−α_{E}, −α_{B} + α_{D} − α_{E}, −α_{B}, α_{C} + α_{E}, α_{B}, 0, 0, 0, α_{C} + α_{D} + α_{H},
α_{A} + α_{B} + α_{C}),
109  (−α_{A} − α_{E}, α_{D} − α_{E}, α_{A}, α_{C} + α_{E}, 0, 0, 0, 0, α_{B} + α_{C} + α_{D} + α_{H},
α_{B} + α_{C}),
110  (−α_{D} − α_{E}, α_{D} − α_{E}, 0, α_{C} + α_{E}, 0, α_{D}, 0, 0, α_{B} + α_{C} + α_{H},
α_{A} + α_{B} + α_{C}),
111  (−α_{A} − α_{D} − α_{E}, α_{D}, α_{A}, α_{C}, 0, α_{D} + α_{E}, 0, 0, α_{B} + α_{C} + α_{H}, α_{B} + α_{C}),
112  (−α_{E}, α_{D} − α_{E}, 0, α_{C} + α_{E}, −α_{B} − α_{C} − α_{D} − α_{H}, 0, α_{B} + α_{C} + α_{D} + α_{H},

α_{B} + α_{C} + α_{D} + α_{H}, 0, α_{A} + α_{B} + α_{C}),

113  (−α_{E}, −α_{B} + α_{D} − α_{E} − α_{H}, 0, α_{C} + α_{E}, 0, 0, α_{B} + α_{H}, 0, α_{C} + α_{D},
α_{A} + α_{B} + α_{C}),
114  (−α_{E}, α_{D} − α_{E}, 0, α_{C} + α_{E}, 0, −α_{A} − α_{B} − α_{C}, 0, α_{A} + α_{B} + α_{C},

α_{B} + α_{C} + α_{D} + α_{H}, 0),

115  (0, −α_{B} + α_{D} − α_{E} − α_{H}, 0, α_{C}, 0, −α_{A} − α_{B} − α_{C}, α_{B} + α_{E} + α_{H},

α_{A} + α_{B} + α_{C}, α_{C} + α_{D}, 0),

116  (0, α_{D}, 0, α_{C}, −α_{B} − α_{C} − α_{D} − α_{E} − α_{H}, −α_{A} − α_{B} − α_{C},

α_{B} + α_{C} + α_{D} + α_{E} + α_{H}, α_{A} + 2α_{B} + 2α_{C} + α_{D} + α_{E} + α_{H}, 0, 0),

117  (0, −α_{B} + α_{D} − α_{E} − α_{H}, −α_{B} − α_{E}, α_{C}, α_{B} + α_{E}, 0, α_{H}, 0, α_{C} + α_{D},
α_{A} + α_{B} + α_{C}),
118  (−α_{A}, −α_{B} + α_{D} − α_{E} − α_{H}, α_{A}, α_{C}, 0, 0, α_{B} + α_{E} + α_{H}, 0, α_{C} + α_{D},
α_{B} + α_{C}),
119  (0, −α_{B} + α_{D} − α_{E} − α_{H}, 0, α_{C}, −α_{C} − α_{D}, 0, α_{B} + α_{C} + α_{D} + α_{E} + α_{H},

α_{C} + α_{D}, 0, α_{A} + α_{B} + α_{C}),

120  (0, −α_{B} + α_{D} − α_{E} − α_{H}, 0, α_{C} − α_{D}, 0, α_{D}, α_{B} + α_{D} + α_{E} + α_{H}, 0,
α_{C}, α_{A} + α_{B} + α_{C}),
121  (0, α_{D}, 0, −α_{B} + α_{C} − α_{D} − α_{E} − α_{H}, −α_{C}, α_{B} + α_{D} + α_{E} + α_{H},

α_{B} + α_{C} + α_{D} + α_{E} + α_{H}, α_{C}, 0, α_{A} + α_{B} + α_{C}),

122  (0, α_{D}, −α_{B} − α_{C} − α_{D} − α_{E}, α_{C}, −α_{H}, 0, α_{H}, α_{B} + α_{C} + α_{D} + α_{E} + α_{H},
0, α_{A} + α_{B} + α_{C}),
123  (−α_{A}, α_{D}, α_{A}, α_{C}, −α_{B} − α_{C} − α_{D} − α_{E} − α_{H}, 0, α_{B} + α_{C} + α_{D} + α_{E} + α_{H},

α_{B} + α_{C} + α_{D} + α_{E} + α_{H}, 0, α_{B} + α_{C}),

124  (0, α_{D}, −α_{B} − α_{C} − α_{D} − α_{E}, α_{C}, 0, −α_{A} − α_{B} − α_{C}, 0,

α_{A} + 2α_{B} + 2α_{C} + α_{D} + α_{E}, α_{H}, 0),

125  (0, α_{D}, 0, −α_{A} − α_{B} + α_{C}, α_{A} + α_{B}, −α_{C}, 0, α_{C},

α_{B} + α_{C} + α_{D} + α_{E} + α_{H}, 0),

126  (−α_{A}, α_{D}, α_{A}, α_{C}, 0, −α_{B} − α_{C}, 0, α_{B} + α_{C}, α_{B} + α_{C} + α_{D} + α_{E} + α_{H}, 0),
127  (−α_{A}, α_{D}, α_{A}, −α_{B} + α_{C}, α_{B}, 0, 0, 0, α_{B} + α_{C} + α_{D} + α_{E} + α_{H}, α_{C}),
128  (−α_{D} − α_{E}, α_{D}, 0, α_{C}, 0, α_{D} + α_{E}, 0, 0, α_{B} + α_{C} + α_{H}, α_{A} + α_{B} + α_{C}),
129  (0, α_{D}, 0, −α_{B} + α_{C} − α_{D} − α_{E} − α_{H}, 0, α_{B} + α_{D} + α_{E} + α_{H},

α_{B} + α_{D} + α_{E} + α_{H}, 0, α_{C}, α_{A} + α_{B} + α_{C}),

130  (0, α_{D}, 0, −α_{A} − α_{B} + α_{C}, α_{A} + α_{B}, 0, 0, 0, α_{B} + α_{C} + α_{D} + α_{E} + α_{H},
α_{C}),
131  (0, −α_{B} + α_{D} − α_{E}, −α_{B} − α_{E}, α_{C}, α_{B} + α_{E}, 0, 0, 0, α_{C} + α_{D} + α_{H},
α_{A} + α_{B} + α_{C}),
132  (0, α_{D}, −α_{B} − α_{C} − α_{D} − α_{E}, α_{C}, 0, 0, 0, α_{B} + α_{C} + α_{D} + α_{E}, α_{H},

α_{A} + α_{B} + α_{C}),
133  (−α_{A} − α_{B} − α_{C} − α_{D} − α_{E}, α_{D}, α_{A}, −α_{B} + α_{C} − α_{H}, α_{B},

α_{B} + α_{D} + α_{E} + α_{H}, α_{H}, α_{C}, 0, 0),

134  (−α_{A} − α_{B} − α_{C} − α_{D} − α_{E}, −α_{B} + α_{D} − α_{E}, α_{A}, α_{C} + α_{E}, α_{B}, α_{D}, 0, α_{C},
α_{H}, 0),
135  (−α_{A} − α_{B} − α_{D} − α_{E}, −α_{B} + α_{D} − α_{E} − α_{H}, α_{A}, α_{C} + α_{E}, α_{B}, α_{D}, α_{H}, 0,
α_{C}, α_{C}),
136  (−α_{D} − α_{E}, −α_{B} + α_{D} − α_{E} − α_{H}, −α_{B} − α_{C}, α_{C} + α_{E}, α_{B}, α_{D}, α_{H}, α_{C}, 0,
α_{A} + α_{B} + α_{C}),
137  (0, −α_{B} + α_{D} − α_{E} − α_{H}, −α_{B} − α_{C} − α_{D} − α_{E}, −α_{A} − α_{B} − α_{D},

α_{A} + 2α_{B} + α_{C} + α_{D} + α_{E}, α_{D}, α_{H}, α_{C}, 0, 0),

138  (−α_{E}, α_{D} − α_{E}, 0, α_{C} + α_{E}, 0, 0, 0, 0, α_{B} + α_{C} + α_{D} + α_{H},
α_{A} + α_{B} + α_{C}),
139  (0, −α_{B} + α_{D} − α_{E} − α_{H}, 0, α_{C}, 0, 0, α_{B} + α_{E} + α_{H}, 0, α_{C} + α_{D},
α_{A} + α_{B} + α_{C}),
140  (0, α_{D}, 0, α_{C}, −α_{B} − α_{C} − α_{D} − α_{E} − α_{H}, 0, α_{B} + α_{C} + α_{D} + α_{E} + α_{H},

α_{B} + α_{C} + α_{D} + α_{E} + α_{H}, 0, α_{A} + α_{B} + α_{C}),

141  (0, α_{D}, 0, α_{C}, 0, −α_{A} − α_{B} − α_{C}, 0, α_{A} + α_{B} + α_{C},

α_{B} + α_{C} + α_{D} + α_{E} + α_{H}, 0),

142  (−α_{A}, α_{D}, α_{A}, α_{C}, 0, 0, 0, 0, α_{B} + α_{C} + α_{D} + α_{E} + α_{H}, α_{B} + α_{C}),
143  (−α_{A} − α_{B} − α_{C} − α_{D} − α_{E}, −α_{B} + α_{D} − α_{E} − α_{H}, α_{A}, α_{C} + α_{E}, α_{B}, α_{D}, α_{H},
α_{C}, 0, 0),
144  (0, α_{D}, 0, α_{C}, 0, 0, 0, 0, α_{B} + α_{C} + α_{D} + α_{E} + α_{H}, α_{A} + α_{B} + α_{C}).

We note that not all the regions of linearity are defined by the same number of inequalities. There are 62 regions (1 to 62) defined by 6 inequalities, 70 regions (63 to 132) defined by 7 inequalities, 10 regions defined by 8 inequalities (133 to 142) and 2 regions (143 to 144) defined by 11 inequalities.

We note in particular that the number (62) of regions defined by the minimum number of inequalities is equal to the number of equivalence classes of reduced expressions for w_{0}. We shall show that there is in fact a natural bijection between these two sets.

Now the vectors which parametrize the PBW-basis elements of U^{−} have all coordinates in ℕ. Thus we shall add the inequalities

a ≥ 0, b ≥ 0, c ≥ 0, d ≥ 0, e ≥ 0, f ≥ 0, g ≥ 0, h ≥ 0, i ≥ 0, j ≥ 0

to those of Table III.

We define a simplicial region to be the subset of ℝ^{10} defined by one of the sets of 6 inequalities in Table III together with the above inequalities asserting that all coordinates are nonnegative. Of these 10 latter inequalities there will be 4 from which the remaining 6 follow using the 6 inequalities from Table III. Thus each simplicial region will have 10 walls, 6 of which are given by Table III and the remaining 4 will have form that some coordinate is greater than or equal to zero. The 62 simplicial regions together with their walls are shown in Table V (where, for example, abgh indicates the four walls a ≥ 0, b ≥ 0, g ≥ 0 and h ≥ 0).

Table V The simplicial regions and their walls.

Region  Main walls  Coordinate walls
1  [A, BC, CD, E, BH] ∥ [BCD]  abgh
2  [A, BC, E, BH] ∥ [B, CD]  abgh
3  [ABC, CD, E, BH] ∥ [A, BCD]  begh
4  [A, BC, E, BCDH] ∥ [BCD, BH]  abgh
5  [A, CD, E, BH] ∥ [BC, D]  abgh
6  [A, BC, CD, BEH] ∥ [BCD, E]  afgh
7  [A, BC, BEH] ∥ [B, CD, E]  afgh
8  [ABC, E, BH] ∥ [A, B, CD]  begh
9  [ABC, E, BCDH] ∥ [A, BCD, BH]  begh
10  [A, E, BCDH] ∥ [BC, D, BH]  abgh

Region Main walls  Coordinate walls
11  [A, CD, BEH] ∥ [BC, D, E]  afgh
12  [A, B, C, BEH] ∥ [CD, BE]  acfh
13  [B, ABC, E, H] ∥ [AB, CD]  bceh
14  [ABC, BCD, E, H] ∥ [ABCD, BH]  begh
15  [A, D, E, BCH] ∥ [BC, BH]  abdg
16  [A, C, D, BEH] ∥ [BC, DE]  adfg
17  [A, B, BEH] ∥ [C, D, BE]  achj
18  [AB, C, BEH] ∥ [A, CD, BE]  cefh
19  [ABC, BE, H] ∥ [AB, CD, E]  cefh
20  [B, ABC, E] ∥ [AB, CD, H]  bchi
21  [ABC, BCD, E] ∥ [B, ABCD, H]  bghi
22  [BCD, E, H] ∥ [ABC, D, BH]  begh
23  [D, E, BCH] ∥ [A, BC, BH]  bdeg
24  [A, DE, BCH] ∥ [BC, E, BH]  adfg
25  [A, C, BDEH] ∥ [BC, DE, BEH]  adfg
26  [A, D, BEH] ∥ [B, C, DE]  adgj
27  [A, B, D, BEH] ∥ [C, BDE]  acdj
28  [AB, C, BE, H] ∥ [CD, ABE]  cefh
29  [B, ABC, CD, E] ∥ [ABCD, H]  bchi
30  [BC, D, E, H] ∥ [ABC, BH]  bdeg
31  [A, C, DE, BH] ∥ [BC, BEH]  adfg
32  [A, BDEH] ∥ [B, C, DE, BEH]  adgj
33  [AB, BEH] ∥ [A, C, D, BE]  cehj

34  [ABC, BE] ∥ [AB, CD, E, H]  cfhi
35  [BCD, E] ∥ [B, ABC, D, H]  bghi
36  [DE, BCH] ∥ [A, BC, E, BH]  defg
37  [C, DE, BH] ∥ [A, BC, BEH]  defg
38  [A, DE, BH] ∥ [B, C, BEH]  adgj
39  [A, B, BDEH] ∥ [C, BDE, BEH]  acdj
40  [AB, D, BEH] ∥ [A, C, BDE]  cdej
41  [AB, BE, H] ∥ [C, D, ABE]  cehj
42  [AB, C, BE] ∥ [CD, ABE, H]  cfhi
43  [ABC, CD, BE] ∥ [ABCD, E, H]  cfhi
44  [B, CD, E] ∥ [ABC, D, H]  bchi
45  [BC, D, E] ∥ [B, ABC, H]  bdgi
46  [BC, DE, H] ∥ [ABC, E, BH]  defg
47  [DE, BH] ∥ [A, B, C, BEH]  degj
48  [AB, BDEH] ∥ [A, C, BDE, BEH]  cdej
49  [AB, BE] ∥ [C, D, ABE, H]  chij
50  [CD, BE] ∥ [ABC, D, E, H]  cfhi
51  [BC, DE] ∥ [B, ABC, E, H]  dfgi
52  [B, DE, H] ∥ [AB, C, BEH]  cdej
53  [AB, BDE, H] ∥ [C, ABDE, BEH]  cdej
54  [AB, D, BE] ∥ [C, ABDE, H]  cdij
55  [C, D, BE] ∥ [ABC, DE, H]  cdfi
56  [B, C, DE] ∥ [ABC, BE, H]  cdfi

57  [B, DE] ∥ [AB, C, BE, H]  cdij
58  [BDE, H] ∥ [AB, C, DE, BEH]  cdej
59  [AB, BDE] ∥ [C, BE, ABDE, H]  cdij
60  [D, BE] ∥ [AB, C, DE, H]  cdij
61  [C, BDE] ∥ [ABC, BE, DE, H]  cdfi
62  [BDE] ∥ [AB, C, BE, DE, H]  cdij

In order to define a natural bijection between the set of equivalence relations of reduced words for w_{0} and the set of simplicial regions we consider the transforms under S_{i}^{j} of the spanning vectors of C_{i} considered in §4. We have one such vector v_{P} for each partial quiver P and one vector v_{j} for each j = 1, 2, 3, 4. These vectors v_{P} , v_{j} can, for example, be calculated by the rectangle algorithm described in §4. These vectors are listed in Table VI.

We consider the incidence properties relating these vectors v_{P} and v_{j} to the walls of the simplicial regions. There are altogether 32 hyperplanes in ℝ^{10} which arise as walls of simplicial regions. 22 of these are hyperplanes appearing in Table III and 10 are the hyperplanes obtained by putting one coordinate equal to 0. There are 22 vectors of form v_{P} and 4 vectors of form v_{j}. These are obtained from the rectangle algorithm which associates with each partial quiver P or each j ∈ {1, 2, 3, 4} a set of positive roots. Such a set of positive roots is translated into a vector with components 0 or 1 by means of the ordering

(α_{1}, α_{3}, α_{1} + α_{2} + α_{3}, α_{3} + α_{4}, α_{2} + α_{3}, α_{1} + α_{2} + α_{3} + α_{4}, α_{2} + α_{3} + α_{4}, α_{1} + α_{2}, α_{4}, α_{2})

determined by the reduced word j = (1, 3, 2, 4, 1, 3, 2, 4, 1, 3). In Table VII we indicate which vectors v_{P} and v_{j} lie on which walls of simplicial regions. A cross in a particular position indicates that the given vector does not lie on the given wall.

Table VI Vectors obtained by the rectangle algorithm.

Partial quiver P  Vector v_{P}
LLL  (1,0,0,0,0,0,0,0,0,0)
LLR  (0,0,1,0,0,0,0,0,1,0)
LRL  (1,1,0,0,0,0,1,1,0,0)
RLL  (1,0,0,1,0,0,0,0,0,1)

Partial quiver P  Vector v_{P}
LRR  (0,1,0,0,0,0,0,1,1,0)
RLR  (0,0,1,1,0,0,0,0,1,1)
RRL  (1,0,0,0,0,0,1,0,0,0)
RRR  (0,0,0,0,0,0,0,0,1,0)
LL-  (0,0,0,0,0,0,0,1,0,0)
LR-  (0,1,0,0,0,1,0,0,0,0)
RL-  (0,0,0,0,0,1,0,0,0,1)
RR-  (0,0,0,1,0,0,0,0,0,0)
- LL  (1,0,0,0,0,0,0,0,0,1)
- LR  (0,0,1,0,0,0,0,0,1,1)
- RL  (1,1,0,0,0,0,1,0,0,0)
- RR  (0,1,0,0,0,0,0,0,1,0)
L- -  (0,1,0,0,0,0,0,1,0,0)
R- -  (0,0,0,1,0,0,0,0,0,1)
- L-  (0,0,1,0,0,0,0,0,0,1)
- R-  (0,1,0,0,0,0,1,0,0,0)
- - L  (1,0,0,0,1,0,0,0,0,0)
- - R  (0,0,0,0,1,0,0,0,1,0)

Integer j  Vector v_{j}

1  (1,0,0,0,1,0,0,0,1,0)
2  (0,0,1,0,0,0,1,0,0,0)
3  (0,1,0,0,0,1,0,0,0,1)
4  (0,0,0,1,0,0,0,1,0,0)

Table VII Incidence table: spanning vectors and walls of simplicial regions. x indicates that the vector lies on the given wall.

L L L R L R R R L L R R - - - - L R - - - -
L L R L R L R R L R L R L L R R - - L R - - 1 2 3 4
L R L L R R L R - - - - L R L R - - - - L R
A  . x . . x x . x x x x x . x . x x x x x x . x x x x
B  x . . x x . . x x x x x x . . x x x . . x x x x x x
C  x x x . x . x x x . x x . . x x x . . x x x x x x x
D  x x …. x x . x x . x x x x . . x x x x x x x x
E  x x . x . x x x x x . x x x … x x . x x x x x x
H  x . x x . . x . x x x x x . x . x x x x . x x x x x
AB  . . x . x . x x x x x x . . x x x x . . x . x x x x
BC x … x x . x x . x x . x . x x . x . x x x x x x
CD x x . x . x x x . . x … x x . x . x x x x x x x
DE x x x . x . x x . x . . x x . . x . x . x x x x x x
BE x . x x … x x x . x x . x . . x . x x x x x x x
BH x x . x . x . . x x x x x x . . x x … x x x x x
ABC . . x x x x x x x . x x x x x x x . x . x . x x x x
BCD x . . x … x . . x . . x . x . x x . x x x x x x
ABE . . . . . . x x x x . x . . . . . x . x x . x x x x
BDE x … x . . x . x . . x . x . x . . x x x x x x x
BCH x x . . . . . . x . x x …. x . x . . x x x x x
BEH x x x x x x . . x x . x x x x x . x . x . x x x x x
ABCD . . . . . . x x . . x . x x x x . x x . x . x x x x
ABDE . . x . x . x x . x . . . . . . x . . x x . x x x x
BCDH x x . x . x …. x . . . . . . x x . . x x x x x
BDEH x x . . . . . . . x . . x x x x x . . x . x x x x x
a  . x . . x x . x x x x x . x . x x x x x . x . x x x
b  x x . x . x x x x . x x x x … x x . x x x x . x
c  x . x x x . x x x x x x x . x x x x . x x x x . x x
d  x x x . x . x x x x x . x x x x x . x x x x x x x .
e  x x x x x x x x x x x x x x x x x x x x … x x x
f  x x x x x x x x x . . x x x x x x x x x x x x x . x
g  x x . x x x . x x x x x x x . x x x x . x x x . x x
h  x x . x . x x x . x x x x x x x . x x x x x x x x .
i  x . x x . . x . x x x x x . x . x x x x x . . x x x
j  x x x . x . x x x x . x . . x x x . . x x x x x . x

The incidence table enables us to describe a bijection between equivalence classes of reduced words and simplicial regions. Let W be the set of hyperplanes in ℝ^{10} which arise as walls of simplicial regions. We have seen that |W| = 22 + 10 = 32.

Proposition 5.1 Let i be a reduced expression for w_{0}. Let P(i) be the set of 6 partial quivers obtained from i using the chamber sets as in §3. Consider the 10 vectors {v_{P} , P ∈ P(i); v_{j}, j = 1, 2, 3, 4}. For each vector in the set consider the set of 9 remaining vectors. Then there is a unique hyperplane W ∈ W such that these 9 vectors lie on W. Moreover, the set of 10 elements of W obtained in this way are the 10 boundary walls of a simplicial region X_{i}. The original 10 vectors v_{P} , P ∈ P(i); v_{j}, j = 1, 2, 3, 4 all lie in X_{i}. Also, the map i → X_{i} gives a bijection between equivalence classes of reduced words and simplicial regions.

The bijection i → X_{i} is described in Table VIII.

Corollary 5.2 The vectors v_{P} , P ∈ P(i); v_{j}, j = 1, 2, 3, 4 are spanning vectors of the region X_{i}.

Proof: This follows from the fact that each vector lies in X_{i} and that each wall of X_{i} contains 9 of the 10 vectors. □

Table VIII The correspondence between reduced words and simplicial regions.

Simplicial region  Reduced word

1  [1, 3, 2, 1, 3, 4, 3, 2, 1, 3]
2  [1, 3, 2, 1, 3, 4, 2, 3, 2, 1]
3  [3, 2, 1, 2, 3, 4, 3, 2, 1, 3]
4  [1, 3, 2, 4, 1, 3, 2, 4, 1, 3]
5  [1, 3, 2, 3, 4, 3, 2, 1, 2, 3]
6  [1, 2, 3, 2, 1, 4, 3, 2, 1, 3]
7  [1, 2, 3, 2, 1, 4, 2, 3, 2, 1]
8  [3, 2, 1, 2, 3, 4, 2, 3, 2, 1]
9  [3, 2, 1, 2, 4, 3, 2, 4, 1, 3]

Simplicial region  Reduced word

10  [1, 3, 2, 4, 3, 2, 1, 2, 4, 3]
11  [1, 2, 3, 2, 4, 3, 2, 1, 2, 3]
12  [1, 2, 3, 1, 2, 1, 4, 3, 2, 1]
13  [3, 2, 1, 3, 2, 3, 4, 3, 2, 1]
14  [3, 2, 1, 4, 3, 2, 3, 4, 1, 3]
15  [1, 3, 4, 3, 2, 3, 1, 2, 4, 3]
16  [1, 2, 3, 4, 3, 2, 3, 1, 2, 3]
17  [1, 2, 3, 1, 2, 4, 3, 2, 1, 2]
18  [2, 1, 2, 3, 2, 1, 4, 3, 2, 1]
19  [2, 3, 2, 1, 2, 3, 4, 3, 2, 1]
20  [3, 2, 1, 3, 4, 2, 3, 2, 4, 1]
21  [3, 2, 1, 4, 3, 2, 4, 3, 4, 1]
22  [3, 2, 4, 3, 2, 1, 2, 3, 4, 3]
23  [3, 4, 3, 2, 1, 2, 3, 2, 4, 3]
24  [1, 4, 3, 4, 2, 3, 1, 2, 4, 3]
25  [1, 2, 4, 3, 4, 2, 3, 1, 2, 3]
26  [1, 2, 3, 4, 3, 2, 1, 2, 3, 2]
27  [1, 2, 3, 4, 3, 1, 2, 1, 3, 2]
28  [2, 3, 1, 2, 1, 3, 4, 3, 2, 1]
29  [3, 2, 1, 3, 4, 3, 2, 3, 4, 1]
30  [3, 4, 3, 2, 3, 1, 2, 3, 4, 3]
31  [1, 4, 3, 2, 3, 4, 3, 1, 2, 3]

32  [1, 2, 4, 3, 4, 2, 1, 2, 3, 2]
33  [2, 1, 2, 3, 2, 4, 3, 2, 1, 2]
34  [2, 3, 2, 1, 4, 2, 3, 2, 4, 1]
35  [3, 2, 4, 3, 2, 1, 2, 4, 3, 4]
36  [4, 3, 4, 2, 1, 2, 3, 2, 4, 3]
37  [4, 3, 2, 1, 2, 3, 4, 3, 2, 3]
38  [1, 4, 3, 2, 3, 4, 1, 2, 3, 2]
39  [1, 2, 4, 3, 4, 1, 2, 1, 3, 2]
40  [2, 1, 2, 3, 4, 3, 2, 1, 3, 2]
41  [2, 3, 1, 2, 3, 4, 3, 2, 1, 2]
42  [2, 3, 1, 2, 1, 4, 3, 2, 4, 1]
43  [2, 3, 2, 1, 4, 3, 2, 3, 4, 1]
44  [3, 2, 3, 4, 3, 2, 1, 2, 3, 4]
45  [3, 4, 3, 2, 3, 1, 2, 4, 3, 4]
46  [4, 3, 4, 2, 3, 1, 2, 3, 4, 3]
47  [4, 3, 2, 1, 2, 3, 4, 2, 3, 2]
48  [2, 1, 2, 4, 3, 4, 2, 1, 3, 2]
49  [2, 3, 1, 2, 4, 3, 2, 1, 2, 4]
50  [2, 3, 2, 4, 3, 2, 1, 2, 3, 4]
51  [4, 3, 4, 2, 3, 1, 2, 4, 3, 4]
52  [4, 3, 2, 1, 3, 2, 3, 4, 3, 2]
53  [2, 1, 4, 3, 2, 3, 4, 1, 3, 2]

54  [2, 3, 4, 3, 1, 2, 1, 3, 2, 4]
55  [2, 3, 4, 3, 2, 3, 1, 2, 3, 4]
56  [4, 3, 2, 3, 4, 3, 1, 2, 3, 4]
57  [4, 3, 2, 3, 4, 1, 2, 3, 2, 4]
58  [4, 2, 3, 2, 1, 2, 3, 4, 3, 2]
59  [4, 2, 3, 4, 1, 2, 1, 3, 2, 4]
60  [2, 3, 4, 3, 2, 1, 2, 3, 2, 4]
61  [4, 2, 3, 2, 4, 3, 1, 2, 3, 4]
62  [4, 2, 3, 2, 4, 1, 2, 3, 2, 4]

6 The transition function S_{i}^{j} on the Lusztig cone C_{i}

We shall now describe an algorithm for calculating S_{i}^{j}(a) for a ∈ C_{i}. As before j is the reduced word (1, 3, 2, 4, 1, 3, 2, 4, 1, 3). The algorithm is best indicated by an example. We illustrate in the case when i = (3, 2, 1, 4, 3, 2, 3, 4, 1, 3). The Lusztig cone for this reduced word i is given by

C_{i} = {a ∈ ℤ^{10} : P_{i}a ≥ 0},

where

P_{i} = ( \begin{matrix}−1 & 1 & 0 & 1 & −1 & 0 & 0 & 0 & 0 & 0 \\ 0 & −1 & 1 & 0 & 1 & −1 & 0 & 0 & 0 & 0 \\ 0 & 0 & −1 & 0 & 0 & 1 & 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −1 & 1 & 0 & 1 & −1 & 0 & 0 \\ 0 & 0 & 0 & 0 & −1 & 1 & −1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & −1 & 1 & 0 & −1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{matrix} ) .

Thus a ∈ C_{i} if and only if a = ac_{1} + bc_{2} + cc_{3} + dc_{4} + ec_{5} + fc_{6} + gc_{7} + hc_{8} + ic_{9} + jc_{10}, where c_{1}, c_{2}, … c_{10} are the columns of Q_{i} = P_{i}^{−1} and a ≥ 0, b ≥ 0, …, j ≥ 0. We have

Q_{i} = ( \begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 0 \\ 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \end{matrix} ) .

Thus a typical point a ∈ C_{i} is given by

a = (i, a + d + f + h + i, a + b + d + e + f + g + h + i, a + b + c + i + j,
a + b + c + d + f + h + i + j, a + b + c + d + e + f + g + h + i + j, g, f + g + h, j, h),

where a ≥ 0, b ≥ 0, …, j ≥ 0.

We now consider S_{i}^{j}(a). We recall that S_{i}^{j} = ϕ_{j}ψ_{i}^{−1} where ψ_{i} : B → Y_{i} and ϕ_{j} : B → ℕ^{k} are bijections. Let ψ_{i}^{−1}(a) = b and ϕ_{i}^{−1}(b) = c. Then S_{i}^{j}(a) = c. We also recall that

b ≡ F̃_{i_{1}}^{a_{1}} F̃_{i_{2}}^{a_{2}} ⋯ F̃_{i_{k}}^{a_{k}} · 1 mod vL′, and
b ≡ F_{j}^{c} mod vL,

where L is the ℤ[v]-lattice spanned by B and L′ is the A-lattice spanned by B. Since ℤ[v] ⊆ A we have L ⊆ L′. Thus

F̃_{i_{1}}^{a_{1}} F̃_{i_{2}}^{a_{2}} ⋯ F̃_{i_{k}}^{a_{k}} · 1 ≡ F_{j}^{c} mod vL′.

We shall describe an algorithm for obtaining the vector c determined by this condition. This is based on the fact that, since the reduced expression j is adapted to the quiver Q_{1} in Figure 11, (in the sense of [8]), the actions of F̃_{1} and F̃_{3} on elements F_{j}^{c}, taken mod vL′, are given by a simple formula. By [9, Cor. 2.5] we have

F̃_{1}^{x}F_{j}^{c} ≡ F_{j}^{c′} mod vL′,
F̃_{3}^{x}F_{j}^{c} ≡ F_{j}^{c′′} mod vL′,

where

c′ = c + (x, 0, 0, 0, 0, 0, 0, 0, 0, 0),
c′′ = c + (0, x, 0, 0, 0, 0, 0, 0, 0, 0).

Similarly, the reduced expression j′ is adapted to the quiver Q_{2} in Figure 11, so the actions of F̃_{2} and F̃_{4} on elements F_{j′}^{c}, taken mod vL′, are given by

F̃_{2}^{x}F_{j′}^{c} ≡ F_{j′}^{c′} mod vL′,
F̃_{4}^{x}F_{j′}^{c} ≡ F_{j′}^{c′′} mod vL′,

where

c′ = c + (x, 0, 0, 0, 0, 0, 0, 0, 0, 0),
c′′ = c + (0, x, 0, 0, 0, 0, 0, 0, 0, 0).

Figure 11: Two quivers of type A_{4}. Q_{1}  Q_{2}^{∘}  ^{∘}  ^{∘}  ^{∘}^{1}  ^{2}  ^{3}  ^{4}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{∘}  ^{∘}  ^{∘}  ^{∘}^{1}  ^{2}  ^{3}  ^{4}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

In order to calculate F̃_{3}^{a_{1}} F̃_{2}^{a_{2}} F̃_{1}^{a_{3}} F̃_{4}^{a_{4}} F̃_{3}^{a_{5}} F̃_{2}^{a_{6}} F̃_{3}^{a_{7}} F̃_{4}^{a_{8}} F̃_{1}^{a_{9}} F̃_{3}^{a_{10}} ·1 mod vL′ we shall write the element being acted on at each stage in the form F_{j}^{c} or F_{j′}^{c′} as appropriate, in order to be able to use the above formulae. The elements can be transformed from form F_{j}^{c} to F_{j′}^{c′} by using the transition function R = R_{j}^{j′} and from form F_{j′}^{c′} to F_{j}^{c} by R^{−1} = R_{j′}^{j}. It follows from the description of the transition function in [8] and the form of j and j′ that R_{j′}^{j} = τR_{j}^{j′}τ where τ = (1 2)(3 4)(5 6)(7 8)(9 10). Thus R^{−1} = τRτ where τ(a, b, c, d, e, f, g, h, i, j) = (b, a, d, c, f, e, h, g, j, i).

We now apply this algorithm to our given example where

(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}, a_{8}, a_{9}, a_{10}) = (i, a + d + f + h + i, a + b + d + e + f + g + h + i, a + b + c + i + j,

a + b + c + d + f + h + i + j, a + b + c + d + e + f + g + h + i + j, g, f + g + h, j, h).

We have

F̃_{3}^{a_{10}} · 1 = F̃_{3}^{a_{10}} · F_{j}^{0} ≡ F_{j}^{c_{1}} mod vL′,

where c_{1} = (0, h, 0, 0, 0, 0, 0, 0, 0, 0). Also,

F̃_{1}^{a_{9}}F_{j}^{c_{1}} ≡ F_{j}^{c_{2}} mod vL′,

where c_{2} = (j, h, 0, 0, 0, 0, 0, 0, 0, 0). One can then check that c_{2} lies in region 144 of §5 and it follows from Table IV that F_{j}^{c_{2}} ≡ F_{j′}^{c_{3}} mod vL′, where c_{3} = (0, 0, 0, 0, 0, 0, 0, 0, h, j). Hence F̃_{4}^{a_{8}}F_{j′}^{c_{3}} ≡ F_{j′}^{c_{4}} mod vL′, where c_{4} = (0, f + g + h, 0, 0, 0, 0, 0, 0, h, j).

We now apply R_{j′}^{j} to c_{4}. Applying τ to this vector we obtain c_{5} = (f + g + h, 0, 0, 0, 0, 0, 0, 0, j, h). We check that this vector lies in region 136 of §5. Hence R_{j}^{j′}(c_{5}) = c_{6}, where c_{6} = (0, j, h, 0, 0, 0, 0, 0, 0, f+g). Applying τ again we obtain c_{7} = (j, 0, 0, h, 0, 0, 0, 0, f + g, 0). Hence

F_{j′}^{c_{4}} ≡ F_{j}^{c_{7}} mod vL′.

Then

F̃_{3}^{a_{7}}F_{j}^{c_{7}} ≡ F_{j}^{c_{8}} mod vL′,

where c_{8} = (j, g, 0, h, 0, 0, 0, 0, f + g, 0). This vector lies in region 139 of §5. Hence

F_{j}^{c_{8}} ≡ F_{j′}^{c_{9}} mod vL′,

where c_{9} = (0, f + h, 0, 0, 0, 0, g, 0, h, j). Then

F̃_{2}^{a_{6}}F_{j′}^{c_{9}} ≡ F_{j′}^{c_{10}} mod vL′,

where c_{10} = (a + b + c + d + e + f + g + h + i + j, f + h, 0, 0, 0, 0, g, 0, h, j).

We now apply R_{j′}^{j} to c_{10}. Applying τ we obtain c_{11} = (f + h, a + b + c + d + e + f + g + h + i + j, 0, 0, 0, 0, 0, g, j, h). This lies in region 144 of §5. Hence R_{j}^{j′}(c_{11}) = c_{12}, where c_{12} = (0, 0, 0, 0, 0, g, j, h, a+ b+c+d+e+f +i, f +g). Applying τ again we obtain c_{13} = (0, 0, 0, 0, g, 0, h, j, f +g, a+b+c+d+e+f +i). Hence

F_{j′}^{c_{10}} ≡ F_{j}^{c_{13}} mod vL′.

Then F̃_{3}^{a_{5}}F_{j}^{c_{13}} ≡ F_{j}^{c_{14}} mod vL′, where c_{14} = (0, a + b + c + d + f + h + i + j, 0, 0, g, 0, h, j, f + g, a + b + c + d + e + f + i). This lies in region 133 of §5. Hence

F_{j}^{c_{14}} ≡ F_{j′}^{c_{15}} mod vL′,

where c_{15} = (e + g, 0, 0, f + h, 0, a + b + c + d + i + j, g, 0, h, j). Thus

F̃_{4}^{a_{4}}F_{j′}^{c_{15}} ≡ F_{j′}^{c_{16}} mod vL′,

where c_{16} = (e + g, a + b + c + i + j, 0, f + h, 0, a + b + c + d + i + j, g, 0, h, j).

We now apply R_{j′}^{j} to c_{16}. Applying τ gives c_{17} = (a+b+c+i+j, e+g, f+h, 0, a+b+c+d+i+j, 0, 0, g, j, h). This lies in region 142 of §5. Hence R_{j}^{j′}(c_{17}) = c_{18}, where c_{18} = (d+f +h, 0, a+b+c+i+j, 0, 0, g, j, h, a+ b + c + d + e + f + i, f + g). Applying τ again gives c_{19} = (0, d + f + h, 0, a + b + c + i + j, g, 0, h, j, f + g, a + b + c + d + e + f + i). Hence

F_{j′}^{c_{16}} ≡ F_{j}^{c_{19}} mod vL′.

We have F̃_{1}^{a_{3}}F_{j}^{c_{19}} ≡ F_{j}^{c_{20}} mod vL′, where c_{20} = (a + b + d + e + f + g + h + i, d + f + h, 0, a + b + c + i + j, g, 0, h, j, f + g, a + b + c + d + e + f + i). This lies in region 104. Hence

F_{j}^{c_{20}} ≡ F_{j′}^{c_{21}} mod vL′,

where c_{21} = (0, a + b + c + i + j, e + g, 0, f + h, c + j, g, a + b + d + i, h, j). Thus

F̃_{2}^{a_{2}}F_{j′}^{c_{21}} ≡ F_{j′}^{c_{22}} mod vL′,

where c_{22} = (a + d + f + h + i, a + b + c + i + j, e + g, 0, f + h, c + j, g, a + b + d + i, h, j).

We now apply R_{j′}^{j} to c_{22}. Applying τ gives c_{23} = (a + b + c + i + j, a + d + f + h + i, 0, e + g, c + j, f + h, a + b + d + i, g, j, h). This lies in region 49. Hence R_{j}^{j′}(c_{23}) = c_{24}, where c_{24} = (0, b + e + g, b + c + j, d + f + h, a + i, g, j, h, a + b + c + d + e + f + i, f + g). Applying τ again gives c_{25} = (b + e + g, 0, d + f + h, b + c + j, g, a + i, h, j, f + g, a + b + c + d + e + f + i). Thus

F_{j′}^{c_{22}} ≡ F_{j}^{c_{25}} mod vL′.

Thus F̃_{3}^{a_{1}}F_{j}^{c_{25}} ≡ F_{j}^{c_{26}} mod vL′, where c_{26} = (b + e + g, i, d + f + h, b + c + j, g, a + i, h, j, f + g, a + b + c + d + e + f + i).

Thus we finally arrive at the conclusion that

F̃_{3}^{a_{1}} F̃_{2}^{a_{2}} F̃_{1}^{a_{3}} F̃_{4}^{a_{4}} F̃_{3}^{a_{5}} F̃_{2}^{a_{6}} F̃_{3}^{a_{7}} F̃_{4}^{a_{8}} F̃_{1}^{a_{9}} F̃_{3}^{a_{10}} · 1 ≡ F_{j}^{c} mod vL′,

where c = (b + e + g, i, d + f + h, b + c + j, g, a + i, h, j, f + g, a + b + c + d + e + f + i).

This procedure can be carried out for each reduced expression i for w_{0}, i.e. for representatives of the 62 commutation classes of such reduced expressions. This was carried out by computer and in each case the coordinates of c are seen to be positive linear combinations of a, b, c, d, e, f, g, h, i, j. Thus we have obtained:

Proposition 6.1 Let i be a reduced expression for w_{0} in type A_{4}. Then S_{i}^{j}(C_{i}) = X_{i}. Moreover the map S_{i}^{j} : C_{i} → X_{i} is linear.

Proof: We know from Corollary 5.2 that S_{i}^{j} maps the spanning vectors of C_{i} to spanning vectors of X_{i}. Since S_{i}^{j} is linear on C_{i} it follows that S_{i}^{j}(C_{i}) = X_{i}. □

7 An Isomorphism of Graphs

We shall now show that the bijection given by i ↦ X_{i} in Proposition 5.1 between equivalence classes of reduced words for w_{0} and simplicial regions is an isomorphism of graphs.

We first introduce a graph structure on the set of equivalence classes of reduced words for w_{0}. We say that two such classes are adjacent if there exist reduced words i, j in these classes such that j can be obtained from i by applying a long braid relation.

For example, the classes containing the reduced words (2, 3, 1, 2, 1, 3, 4, 3, 2, 1), (2, 3, 1, 2, 3, 4, 3, 2, 1, 2) are adjacent. For the former is equivalent to (2, 3, 1, 2, 3, 4, 3, 1, 2, 1), which can be obtained from the latter by applying the long braid relation 212 → 121.

Secondly, we introduce a graph structure on the set of simplicial regions. Two simplicial regions are said to be adjacent if they have a common wall with equation Q = 0 and there is a bijection between their remaining walls such that corresponding walls have equations Q_{i} = 0, Q_{i} + λ_{i}Q = 0 for some λ_{i} ∈ ℝ.

For example, the simplicial regions 28 and 41 are adjacent. We see from Table V that their walls are as follows:

28 [AB, C, BE, H]∥[CD, ABE] cefh
41 [AB, BE, H]∥[C, D, ABE] cehj.

The common wall is taken as C, i.e. α_{C} = 0. The bijection between the remaining walls satisfies CD ↔ D, f ↔ j and X ↔ X for all other walls X. This bijection is of the required form since α_{C}(v) = f − j.

We shall obtain an isomorphism of graphs by showing that i, j are adjacent if and only if X_{i}, X_{j} are adjacent. We first need the following property of a pair of reduced words:

Proposition 7.1 Two classes of reduced words [i], [i′] for w_{0} differ by a simple long braid relation if and only if their two families of chamber sets differ by just one chamber set.

We illustrate this proposition with the reduced words (2, 3, 1, 2, 3, 4, 3, 1, 2, 1) and (2, 3, 1, 2, 3, 4, 3, 2, 1, 2), which differ by a long braid relation 121 → 212. Their respective chamber diagrams are shown in Figure 12, and their respective chamber sets are:

1 2 3 4 5 2 3 1 2 3 4 3 1 2 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}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} 24525 1245125 15  1235  1 2 3 4 5 2 3 1 2 3 4 3 2 1 2_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} 24525 1245125 15  124 Figure 12: Two words differing by a long braid relation.

25, 15, 245, 125, 1245, 1235,

and

25, 15, 245, 125, 1245, 124.

Proof: First suppose that the reduced words i, i′ differ by a single long braid relation. Consider the crossings in their respective chamber diagrams. The only difference in the order of the crossings is in a set of three consecutive crossings. We may assume

i = … k, k + 1, k, …
i′ = … k + 1, k, k + 1, …

where the dotted letters are the same in i, i′. The corresponding crossings have form:

i : … (pq), (pr), (qr) …
i′ : … (qr), (pr), (pq) …

where the dotted crossings are the same for i, i′, and p < q < r. Now the three crossings (pq), (pr), (qr) bound a chamber in each chamber diagram. This chamber has form as shown in Figure 13.

The chamber sets corresponding to this chamber in CD(i) and CD(i′) are different, but all other chamber sets in CD(i), CD(i′) are the same. Thus the families of chamber sets in CD(i), CD(i′) differ by just one chamber set.

Now suppose conversely that the chamber sets in CD(i), CD(i′) differ by just one chamber set. Let C be the chamber in CD(i) whose chamber set does not appear in CD(i). We note that each crossing (ij) appears just once in each chamber diagram and, if i < j, the chamber set for the chamber on the left of (ij) gives the chamber set on the right of (ij) replacing j by i.

pqr ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  in CD(i), pqr _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  in CD(i′). Figure 13: Chamber bounded by (pq), (pr) and (qr).

(a) Suppose first that chambers C, C′ occupy the same positions in their respective chamber diagrams. Let the left and right hand crossings of C be (ij), (kl), respectively, with i < j, k < l. Since all other crossings are the same in the two chamber diagrams, the left and right hand end crossings of C′ must be (kl), (ij), respectively. In CD(i) string k is above string j at crossing (ij), whereas string j is above string k at crossing (kl). Thus the crossing (jk) must be above chamber C in CD(i). A similar argument show that crossing (jk) is below chamber C′ in CD(i′). Since crossing (jk) appears in the same row in both chamber diagrams we obtain a contradiction. Thus C, C′ cannot occupy the same positions in CD(i), CD(i′) respectively.

(b) Consider the sequence of chambers in the row of CD(i) containing C. This sequence has form

… C_{1}, C, C_{2}, …

and their chamber sets have form

… S(C_{1}), S(C), S(C_{2}), ….

When C is removed, we know that S(C_{1}), S(C_{2}) will be consecutive chamber sets in the same row of CD(i′). Thus S(C_{1}), S(C_{2}) differ by a single number, i.e.

|S(C_{1}) ∩ S(C_{2})| = |S(C_{1})| − 1 = |S(C_{2})| − 1.

This implies that the left and right hand end crossings of C must have form

(ij), (jk) for i < j < k,
or (jk), (ij) for i < j < k.

So, in passing from …, C_{1}, C, C_{2}, … to …, C_{1}, C_{2}, …, crossings (ij) and (jk) are removed and crossing (ik) is added. Since the set of crossings in CD(i), CD(i′) is the same, the crossing (ik) must be removed from CD(i) in adjoining C′ and crossings (ij) and (jk) added. Thus the row of CD(i′) containing C′ is the row of CD(i) containing the crossing (ik).

(c) Suppose the left and right end crossings of C are (ij), (jk) respectively with i < j < k. Then the crossing (ik) must be below the level of C in CD(i). Alternatively, if the left and right end crossings of C are (jk), (ij), respectively with i < j < k, the crossing (ik) will be above the level of C in CD(i). These two possibilities are illustrated in Figure 14.

ij  kj^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} (ik)  Case 1 C  jk  ji^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  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kj^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  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ji^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}(ik) C Figure 14: The possibilities for crossing (ik).

(d) We next show that crossing (ik) in CD(i) lies either in the row immediately below that of (ij), (jk), or the row immediately above.

It will be sufficient to consider Case 1 and show that (ik) is in the row below (ij), (jk). A similar argument will work in Case 2. So suppose if possible that (ik) is not in the line immediately below (ij) and (jk). Then there will be a crossing (im) with i < m between (ij) and (ik). Crossing (jm) cannot occur between (ij) and (jk) since then (ij) and (jk) could not bound the same chamber. So there must be a crossing (mk) between (ik) and (jk). Also all crossings involving m between (im) and (mk) are on a level below the level of C. See Figure 15.

ij  kj^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} (im)  (mk) (ik) Figure 15: All crossings involving m between (im) and (mk) are on a level below the level of C.

Now consider the chamber diagram CD(i′). The crossings involving m in CD(i′) are in the same positions as in CD(i) since m ∉ {i, j, k}. Crossing (ik) lies to the right of (im) and to the left of (mk) in CD(i′).

Now string i remains below string m to the right of (im) and string k remains below string m to the left of (mk) in CD(i′). Thus crossing (ik) lies below string m in CD(i′). Since all crossings involving m between (im) and (mk) are on a level below that of C, the level of crossing (ik) in CD(i′) is below the level of C in CD(i). But the level of (ik) in CD(i′) is the same as the level of C in CD(i), so we have a contradiction. Thus crossing (ik) in CD(i) lies on the line below (ij) and (jk). In Case 2 (ik) lies on the line above (ij) and (jk).

(e) We show next there is no crossing involving j between (ij) and (jk) in CD(i). Suppose we are in Case 1. If there is a crossing (mj) with m < j between (ij) and (jk) then crossing (mk) would be to the right of (mj) and the left of (jk). But then (ij) and (jk) could not bound the same chamber. Similarly if there is a crossing (jm) with j < m between (ij) and (jk) then crossing (im) would be to the right of (ij) and to the left of (jm). Then (ij) and (jk) could not bound the same chamber. A similar argument applies in Case 2.

(f) Now we show there is no crossing involving i between (ij) and (ik) and no crossing involving k between (ik) and (jk). Suppose we are in Case 1. Suppose there were a crossing (im), i < m, between (ij) and (ik). Then string m remains below string j and above string i between (im) and (jk). Since string i is in the row below string j at the crossing (ik), by (e), we have a contradiction. Similarly there is no crossing (mk), m < k, between (ik) and (jk). A similar argument holds in Case 2.

(g) We now know that the chambers C and C′ are as in Figure 16.

Case 1:  ij  k  kj i^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} C  in CD(i), ij  k  kj i_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.} C′  in CD(i′).

Case 2:  ij  k  kj i^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} C′  in CD(i′). ij  k  kj i^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} C  in CD(i), Figure 16: The chambers C and C′.

In Case 1 we may choose a total order on the crossings in CD(i) and CD(i′) compatible with the diagrams of the form

… (ij), (ik), (jk), …  in CD(i)
… (jk), (ik), (ij), …  in CD(i′),

where the crossings agree apart from the given triples. The crossings in this order give rise to reduced words

⋯ s_{p}s_{p+1}s_{p} ⋯
⋯ s_{p+1}s_{p}s_{p+1} ⋯

commutation equivalent to i, i′ respectively and differing by a long braid relation. A similar agreement holds in Case 2, so the proposition is proved.

Proposition 7.2 Let i, i′ be reduced words for w_{0}. Their commutation classes [i], [i′] are adjacent if and only if the simplicial regions X_{i}, X_{i′} are adjacent.

Proof: Suppose [i], [i′] are adjacent. Then i, i′ both determine 6 partial quivers and these two sets of 6 have 5 in common, by Proposition 7.1. Let these partial quivers be

P(i) = {P_{1}, P_{2}, P_{3}, P_{4}, P_{5}, P_{6}}, and
P(i′) = {P_{1}, P_{2}, P_{3}, P_{4}, P_{5}, P′_{6}}.

Then, by Proposition 5.1, X_{i} has spanning vectors {v_{P} , P ∈ P(i), v_{1}, v_{2}, v_{3}, v_{4}} and X_{i′} has spanning vectors {v_{P} , P ∈ P(i′), v_{1}, v_{2}, v_{3}, v_{4}}. Let W be the common wall of X_{i}, X_{i′}, spanned by vectors v_{P_{1}}, v_{P_{2}}, v_{P_{3}}, v_{P_{4}}, v_{P_{5}} and v_{1}, v_{2}, v_{3}, v_{4}. Let W_{i}, i = 1, 2, …, 9, be the walls of X_{i} containing all spanning vectors of X_{i} except one, where the omitted spanning vector is not v_{P_{6}}. Let W′_{i}, i = 1, 2, …, 9, be the corresponding walls of X_{i′}. Let W be given by equation Q = 0 and W_{i} by equation Q_{i} = 0. Then we have

W_{i} ∩ W_{i′} = W_{i} ∩ W and
W_{i} ∩ W = {x : Q(x) = 0, Q_{i}(x) = 0}.

Since W_{i} ∩ W ⊆ W′_{i}, the corresponding ideals I(W_{i} ∩ W), I(W′_{i}) satisfy I(W′_{i}) ⊆ I(W_{i} ∩ W). Let W′_{i} have equation Q′_{i} = 0. Then Q′_{i} lies in the ideal generated by Q and Q_{i}. The only linear polynomials in this ideal are those of the form λQ + λ_{i}Q_{i} with λ, λ_{i} ∈ ℝ. Thus Q′_{i} has form λQ + λ_{i}Q_{i}. Now W′_{i} is not equal to W so λ_{i} ≠ 0. Without loss of generality we may choose λ_{i} = 1, so Q′_{i} = Q_{i} + λQ for some λ ∈ ℝ. This shows that the simplicial regions X_{i}, X_{i′} are adjacent.

Conversely, suppose i, i′ are reduced words such that the regions X_{i}, X_{i′} are adjacent. Thus they have a common wall W given by equation Q = 0 and a 1 − 1 correspondence between their remaining walls

W_{i} ↔ W′_{i} i = 1, 2, …, 9,

where W_{i} has equation Q_{i} = 0 and W′_{i} has equation Q_{i} + λ_{i}Q = 0 for λ_{i} ∈ ℝ. Let L_{i}, L′_{i} be the 1-dimensional subspaces of ℝ^{10} given by

L_{i} = ∩_{j,j≠i}W_{j} ∩ W, L′_{i} = ∩_{j,j≠i}W′_{j} ∩ W.

Both L_{i} and L′_{i} are given by the equations Q = 0, Q_{j} = 0, j ∈ {1, 2, …, 9}, j ≠ i. Thus L_{i} = L′_{i}. Hence the regions X_{i}, X_{i′} have 9 common spanning vectors, since a spanning vector has all coordinates 0 or 1 so is determined by the 1-dimensional subspace containing it. Four of these common spanning vectors are the vectors v_{1}, v_{2}, v_{3}, v_{4} of Proposition 5.1. The remaining spanning vectors of X_{i}, X_{i′} have form v_{P}, for P ∈ P(i), P ∈ P(i′) respectively. Since distinct partial quivers P give distinct vectors v_{P} we see that P(i) and P(i′) must have 5 partial quivers in common. Thus by Proposition 7.1 the commutation classes [i] and [i′] of reduced words must be adjacent. □

This proposition shows that the correspondence [i] → X_{i} is an isomorphism of graphs. This graph, with 62 vertices, is shown in Figure 17. There is an involution ι on the set of reduced expressions for w_{0}, taking a reduced expression i = (i_{1}, i_{2}, …, i_{10}) to (5 − i_{1}, 5 − i_{2}, …, 5 − i_{10}) (thus applying the graph automorphism of the Dynkin diagram of type A_{4}). This induces an involution on the set of commutation classes of reduced expressions, and thus on the set of simplicial regions. We have numbered the simplicial regions in such a way that ι takes region m to region 63 − m, for m = 1, 2, … 62. In fact, ι induces an automorphism of the graph of simplicial regions.

8 A Correspondence of Monomials

We shall now show how monomials in the Kashiwara operators given by vectors in a Lusztig cone are related to the corresponding monomials in divided powers in the generators F_{i} of U^{−}. To do this we shall relate both to elements of a basis in U^{−} of PBW-type. For this purpose we shall need to consider root vectors of U^{−}. We follow ideas of Xi in [19].

Let l = (l_{1}, l_{2}, …, l_{k}) be the reduced expression for w_{0} given by

(l_{1}, l_{2}, …, l_{k}) = (n, n − 1, n, n − 2, n − 1, n, …, 1, 2, …, n).

The elements of the corresponding basis B_{l} of PBW-type are:

F_{l}^{c} = F_{l_{1}}^{(c_{1})}T′′_{l_{1},−1}(F_{l_{2}}^{(c_{2})}) ⋯ T′′_{l_{1},−1}T′′_{l_{2},−1} ⋯ T′′_{l_{k−1},−1}(F_{l_{k}}^{(c_{k})}).

The corresponding sequence of positive roots is:

α_{l_{1}}, s_{l_{1}}(α_{l_{2}}), s_{l_{1}}s_{l_{2}}(α_{l_{3}}), …, s_{l_{1}}s_{l_{2}} ⋯ s_{l_{k−1}}(α_{l_{k}}).

We write α_{ij} = α_{i} + ⋯ + α_{j−1} for i < j. For each p with 1 ≤ p ≤ k we have s_{l_{1}}s_{l_{2}} ⋯ s_{l_{p−1}}(α_{l_{p}}) = α_{ij} for some i < j, and each α_{ij} appears in this way for precisely one p. We then define

F_{ij} = T′′_{l_{1},−1}T′′_{l_{2},−1} ⋯ T′′_{l_{p−1},−1}(F_{l_{p}}).

Each F_{ij} is an element of U^{−} of weight −α_{ij}. The elements F_{ij} will be called the root vectors of U^{−}.

Lemma 8.1 We have:

F_{ij} = (−1)^{j−i−1}T_{i}^{−1}T_{i+1}^{−1} ⋯ T_{j−2}^{−1}F_{j−1}

for i < j.

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 ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.} The graph of simplicial regions in Type A_{4} Figure 17

55 Proof: Let p be the positive integer such that s_{l_{1}}s_{l_{2}} ⋯ s_{l_{p−1}}(α_{l_{p}}) = α_{ij}. Then we have:

s_{l_{1}}s_{l_{2}} ⋯ s_{l_{p}} = s_{n}s_{n−1}s_{n} ⋯ s_{i+1}s_{i+2} ⋯ s_{n}s_{i}s_{i+1} ⋯ s_{n+i+1−j}.

It is readily checked that the word

s_{j−2}s_{j−3} ⋯ s_{i}s_{n}s_{n−1}s_{n} ⋯ s_{i+1}s_{i+2} ⋯ s_{n}s_{i}s_{i+1} ⋯ s_{n+i+1−j}

is reduced and that this element of W transforms α_{n+i+1−j} into α_{j−1}. It follows from Lusztig [8, §1.3(c)], that

T′′_{j−2,−1}T′′_{j−3,−1} ⋯ T′′_{i,−1}T′′_{n,−1}T′′_{n−1,−1}T′′_{n,−1} ⋯ T′′_{n+i+1−j}(F_{n+i+1−j}) = F_{j−1}.

This asserts that:

T′′_{j−2,−1}T′′_{j−3,−1} ⋯ T′′_{i,−1}(F_{ij}) = F_{j−1},

and so

F_{ij} = (T′′_{i,−1})^{−1}(T′′_{i+1,−1})^{−1} ⋯ (T′′_{j−2,−1})^{−1}(F_{j−1})
= r_{i}^{−1}T_{i}^{−1}r_{i+1}^{−1}T_{i+1}^{−1} ⋯ r_{j−2}^{−1}T_{j−2}^{−1}(F_{j−1})

since T′′_{i,−1} = T_{i}r_{i}. It follows, using the definition of r_{i}, that

F_{ij} = (−1)^{j−i−1}T_{i}^{−1}T_{i+1}^{−1} ⋯ T_{j−2}^{−1}(F_{j−1}),

as required. □

Lemma 8.2 We have

F_{ij} = F_{i}F_{i+1,j} − vF_{i+1,j}F_{i},

for i + 1 < j.

Proof: First note that

F_{i,i+2} = −T_{i}^{−1}(F_{i+1}) = F_{i}F_{i+1} − vF_{i+1}F_{i}.

Assuming the required result inductively for F_{i′,j′} with j′ − i′ < j − i we have:

F_{ij} = −T_{i}^{−1}F_{i+1,j}
= −T_{i}^{−1}(F_{i+1}F_{i+2,j} − vF_{i+2,j}F_{i+1})
= (F_{i}F_{i+1} − vF_{i+1}F_{i})F_{i+2,j} − vF_{i+2,j}(F_{i}F_{i+1} − vF_{i+1}F_{i})
= F_{i}(F_{i+1}F_{i+2,j} − vF_{i+2,j}F_{i+1}) − v(F_{i+1}F_{i+2,j} − vF_{i+2,j}F_{i+1})F_{i}
= F_{i}F_{i+1,j} − vF_{i+1,j}F_{i}.□

We now obtain commutation relations between root vectors and their divided powers analagous to those obtained by Xi in [19, §5.6]:

Proposition 8.3

(a) F_{pq}^{(M)}F_{rs}^{(N)} = F_{rs}^{(N)}F_{pq}^{(M)}, if q < r or r < p < q < s,
(b) F_{pq}^{(M)}F_{rs}^{(N)} = v^{MN}F_{rs}^{(N)}F_{pq}^{(M)}, if r < p < q = s,
(c) F_{pq}^{(M)}F_{rs}^{(N)} = v^{−MN}F_{rs}^{(N)}F_{pq}^{(M)}, if p = r < q < s or p < r < q = s,
(d) F_{pq}^{(M)}F_{rs}^{(N)} = ∑_{0≤t≤min(M,N)} v^{(M−t)(N−t)}F_{rs}^{(N−t)}F_{ps}^{(t)}F_{pq}^{(M−t)}, if q = r, and
(e) F_{pq}^{(M)}F_{rs}^{(N)} = F_{rs}^{(N)}F_{pq}^{(M)}, if s < p or p < r < s < q.
(f) F_{pq}^{(M)}F_{rs}^{(N)} = ∑_{0≤t≤min(M,N)} v^{− \frac{1}{2} t(t−1)}(v^{−1} − v)^{t}[t]!F_{rq}^{(t)}F_{rs}^{(N−t)}F_{pq}^{(M−t)}F_{ps}^{(t)}, if p < r < q < s.

Note that (e) is a restatement of (a). Using the relations (a)–(e) we may now obtain our main result of this section:

Theorem 8.4 Suppose i is a reduced expression for w_{0}, and suppose a ∈ C_{i}. Then we have

F̃_{i_{1}}^{a_{1}} F̃_{i_{2}}^{a_{2}} ⋯ F̃_{i_{k}}^{a_{k}} · 1 ≡ F_{i_{1}}^{(a_{1})}F_{i_{2}}^{(a_{2})} ⋯ F_{i_{k}}^{(a_{k})} mod vL′.

Proof: Let b = F_{i_{1}}^{(a_{1})}F_{i_{2}}^{(a_{2})} ⋯ F_{i_{k}}^{(a_{k})} ∈ U^{−}. Then it is shown by Marsh in [14, §4.1] that in type A_{4} for a ∈ C_{i} we have b ∈ B. So there is a unique c ∈ ℕ^{k} such that b ≡ F_{l}^{c} mod vL (with l as above).

Since b ∈ L and B_{l} is a ℤ[v]-basis of L, b is a linear combination of elements F_{l}^{d} for d ∈ ℕ^{k}, with coefficients in ℤ[v]. We shall obtain information about this expression for b by means of an algorithm.

This algorithm is as follows. It deals with expressions of the form

∑_{p=(p_{1},p_{2},... ,p_{t})∈P} λ_{p,a}F_{γ_{1}}^{(f_{1})}F_{γ_{2}}^{(f_{2})} ⋯ F_{γ_{u}}^{(f_{u})}

where γ_{1}, γ_{2}, …, γ_{u} are positive roots, P is a subset of ℕ^{t} defined by certain linear inequalities, a = (a_{1}, a_{2}, …, a_{k}) is as above, f_{1}, f_{2}, …, f_{u} are certain linear functions in a_{1}, a_{2}, …, a_{k} and p_{1}, p_{2}, …, p_{t}, and λ_{p,a} ∈ ℤ[v].

The element F_{i_{1}}^{(a_{1})}F_{i_{2}}^{(a_{2})} ⋯ F_{i_{k}}^{(a_{k})} is an expression of this form with just one summand, and the algorithm starts with this expression.

Let α^{p} = s_{l_{1}}s_{l_{2}} ⋯ s_{l_{p−1}}(α_{l_{p}}). Then α^{1}, α^{2}, …, α^{k} is the total ordering on the positive roots determined by l. We write α^{p} < α^{q} if p < q. The aim of the algorithm is to transform F_{i_{1}}^{(a_{1})}F_{i_{2}}^{(a_{2})} ⋯ F_{i_{k}}^{(a_{k})} into an expression of the above general form in which the positive roots appearing satisfy γ_{1} < γ_{2} < ⋯ < γ_{u}. Consider adjacent pairs F_{γ_{i}}^{(f_{i})}F_{γ_{i+1}}^{(f_{i+1})} in each monomial in the sum, where γ_{i} ≠ γ_{i+1}. Find the first adjacent pair for which γ_{i} > γ_{i+1}. Then, by using one of the relations (a)–(e) of Proposition 8.3 it is possible to rewrite F_{γ_{i}}^{(f_{i})}F_{γ_{i+1}}^{(f_{i+1})}. In fact just one of these relations can be used. Applying this procedure to the first adjacent pair in each monomial gives us another expression of the same general form, so that the algorithm can be repeated.

Whenever an adjacent pair F_{γ_{i}}^{(f_{i})}F_{γ_{i+1}}^{(f_{i+1})} is obtained with γ_{i} = γ_{i+1} it is replaced by the equivalent

expression [ \begin{matrix}f_{i} + f_{i+1} \\ f_{i} \end{matrix} ] F_{γ_{i}}^{(f_{i}+f_{i+1})}.

If the algorithm terminates we shall have an expression of b as a ℤ[v]-combination of elements in B_{l}. It is not a priori clear that the algorithm will terminate. However, the algorithm was implemented in Maple for each reduced expression i for w_{0} in type A_{4}, and did in fact terminate in each case. The implementation did not calculate the coefficients λ_{p,a} explicitly, but it did calculate the smallest power of v appearing in each λ_{p,a} (which by construction must appear with coefficient 1). This number is a quadratic expression in p_{1}, p_{2}, …, p_{t}, a_{1}, a_{2}, …, a_{k}.

Next, consider the monomial

F̃_{i_{1}}^{a_{1}} F̃_{i_{2}}^{a_{2}} ⋯ F̃_{i_{k}}^{a_{k}} · 1

in the Kashiwara root operators F̃_{i}. It was shown in §6 that the map S_{i}^{j} is linear on C_{i}, where j = (1, 3, 2, 4, 1, 3, 2, 4, 1, 3) and i is any reduced expression for w_{0}. In fact, a similar calculation, using Maple, shows that S_{i}^{j} is linear on C_{i} for all pairs of reduced expressions i, j. In particular, S_{i}^{l} is linear on C_{i} for all i. It turns out that l appears to be a reduced expression for which the calculation of S_{i}^{l} is as simple as possible. Using Theorem 5.5 in the paper [15] by the second author, we have a description of this map, giving us S_{i}^{l}(a) = c such that

F̃_{i_{1}}^{a_{1}} F̃_{i_{2}}^{a_{2}} ⋯ F̃_{i_{k}}^{a_{k}} · 1 ≡ F_{l}^{c} mod vL′.

The coefficient of F_{l}^{c} in the expression for b obtained by the above algorithm was then checked and seen to lie in 1 + vℤ[v]. It follows from Lusztig [8, §§2.3, 3.2] that the coefficients of all other terms F_{l}^{d} in the sum must lie in vℤ[v], and thus that b ≡ F_{l}^{c} mod vL, and so b ≡ F_{l}^{c} mod vL′ (by [9, §2.3]). It follows that b = F_{i_{1}}^{(a_{1})}F_{i_{2}}^{(a_{2})} ⋯ F_{i_{k}}^{(a_{k})} ≡ F̃_{i_{1}}^{a_{1}} F̃_{i_{2}}^{a_{2}} ⋯ F̃_{i_{k}}^{a_{k}} · 1 mod vL′ as required. □

We now give an example to demonstrate the above proof. We take i = (3, 2, 1, 4, 3, 2, 3, 4, 1, 3), and a ∈ C_{i}. The corresponding monomial is

b = F_{3}^{(a_{1})}F_{2}^{(a_{2})}F_{1}^{(a_{3})}F_{4}^{(a_{4})}F_{3}^{(a_{5})}F_{2}^{(a_{6})}F_{3}^{(a_{7})}F_{4}^{(a_{8})}F_{1}^{(a_{9})}F_{3}^{(a_{10})},

The ordering on the set of positive roots induced by l = (4, 3, 4, 2, 3, 4, 1, 2, 3, 4) is

(α_{45}, α_{35}, α_{34}, α_{25}, α_{24}, α_{23}, α_{15}, α_{14}, α_{13}, α_{12}).

We begin the algorithm by writing b in the form

b = F_{34}^{(a_{1})}F_{23}^{(a_{2})}F_{12}^{(a_{3})}F_{45}^{(a_{4})}F_{34}^{(a_{5})}F_{23}^{(a_{6})}F_{34}^{(a_{7})}F_{45}^{(a_{8})}F_{12}^{(a_{9})}F_{34}^{(a_{10})},

The first adjacent pair of root vectors appearing in the wrong order is F_{12}^{(a_{3})}F_{45}^{(a_{4})}. Applying relation (a) of Proposition 8.3 we obtain

b = F_{34}^{(a_{1})}F_{23}^{(a_{2})}F_{45}^{(a_{4})}F_{12}^{(a_{3})}F_{34}^{(a_{5})}F_{23}^{(a_{6})}F_{34}^{(a_{7})}F_{45}^{(a_{8})}F_{12}^{(a_{9})}F_{34}^{(a_{10})}.

Next apply relation (a) to the adjacent pair F_{23}^{(a_{2})}F_{45}^{(a_{4})} and obtain

b = F_{34}^{(a_{1})}F_{45}^{(a_{4})}F_{23}^{(a_{2})}F_{12}^{(a_{3})}F_{34}^{(a_{5})}F_{23}^{(a_{6})}F_{34}^{(a_{7})}F_{45}^{(a_{8})}F_{12}^{(a_{9})}F_{34}^{(a_{10})}.

Now we apply relation (d) to the adjacent pair F_{34}^{(a_{1})}F_{45}^{(a_{4})} and obtain

b = ∑ v^{(a_{1}−p_{1})(a_{4}−p_{1})}F_{45}^{(a_{4}−p_{1})}F_{35}^{(p_{1})}F_{34}^{(a_{1}−p_{1})}F_{23}^{(a_{2})}F_{12}^{(a_{3})}F_{34}^{(a_{5})}F_{23}^{(a_{6})}F_{34}^{(a_{7})}F_{45}^{(a_{8})}F_{12}^{(a_{9})}F_{34}^{(a_{10})},

where the sum is over those p_{1} satisfying 0 ≤ p_{1} ≤ min(a_{1}, a_{4}). Continuing the algorithm in this way we obtain, after 25 steps,

b = ∑_{(p_{1},p_{2},...p_{10})∈P} λ_{p,a}F_{45}^{(f_{1})}F_{35}^{(f_{2})}F_{34}^{(f_{3})}F_{25}^{(f_{4})}F_{24}^{(f_{5})}F_{23}^{(f_{6})}F_{15}^{(f_{7})}F_{14}^{(f_{8})}F_{13}^{(f_{9})}F_{12}^{(f_{10})},

where

f = (f_{1}, f_{2}, f_{3}, f_{4}, f_{5}, f_{6}, f_{7}, f_{8}, f_{9}, f_{10}) = (a_{4} + a_{8} − p_{1} − p_{6} − p_{7} − p_{8}, p_{1} + p_{8},
a_{1} + a_{5} + a_{7} + a_{10} − p_{1} − p_{2} − p_{4} − p_{5} − p_{8} − p_{9} − p_{10}, p_{7}, p_{2} + p_{5} − p_{7} + p_{10},
a_{2} + a_{6} − p_{2} − p_{3} − p_{5} − p_{10}, p_{6}, p_{4} − p_{6} + p_{9}, p_{3} − p_{4} − p_{9}, a_{3} + a_{9} − p_{3}).

The lowest power of v in λ_{p,a} is xT x^{t} where x = (a_{1}, a_{2}, …, a_{10}, p_{1}, p_{2}, …, p_{10}) and

T = ( \begin{matrix}0 & 0 & 0 & 1 & −1 & 0 & −1 & 1 & 0 & −1 & −1 & 1 & 0 & 1 & 1 & −1 & −1 & −1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & −1 & 1 & 0 & 0 & 1 & 0 & −1 & 1 & −1 & −1 & 0 & 0 & 0 & −1 & −1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & −1 & 0 & 0 & 0 & −1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & −1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & −1 & 1 & 0 & −1 & 0 & 0 & 0 & 1 & 1 & −1 & −1 & −1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & −1 & −1 & 0 & 0 & 0 & −1 & −1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & −1 & 0 & 0 & 0 & 0 & 0 & −1 & −1 & −1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 & 1 & 0 & 1 & −1 & 0 & 1 & 1 & −1 & 0 & −1 & −1 & 1 & 1 & 0 & −1 & −1 \\ 0 & 0 & 0 & 0 & −1 & 1 & −1 & 0 & 0 & −1 & 0 & 1 & −1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & −1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & −1 & 0 & 0 & −1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & −1 \\ 0 & 0 & 0 & 0 & 0 & 0 & −1 & 0 & 0 & −1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & −1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & −1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & −1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & −1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & −1 & −1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & −1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & −1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix} ) .

Here P is the set of (p_{1}, p_{2}, …, p_{10}) ∈ ℕ^{10} satisfying p_{1} ≤ a_{1}, p_{1} ≤ a_{4}, p_{2} ≤ a_{2}, p_{2} ≤ a_{5}, p_{3} ≤ a_{3}, p_{3} ≤ a_{6}, p_{4} ≤ p_{3}, p_{4} ≤ a_{7}, p_{5} ≤ p_{2} − p_{3} + a_{2} + a_{6}, p_{5} ≤ −p_{4} + a_{7}, p_{6} ≤ p_{4}, p_{6} ≤ a_{8}, p_{7} ≤ p_{2} + p_{5}, p_{7} ≤ −p_{6} + a_{8}, p_{8} ≤ −p_{1} − p_{2} − p_{4} − p_{5} + a_{1} + a_{5} + a_{7}, p_{8} ≤ −p_{6} − p_{7} + a_{8}, p_{9} ≤ p_{3} − p_{4}, p_{9} ≤ a_{10}, p_{10} ≤ −p_{2} − p_{3} − p_{5} + a_{2} + a_{6}, p_{10} ≤ −p_{9} + a_{10}.

The function S_{i}^{l}(a) = c is given in this example by

c = (−a_{1} + a_{4}, a_{1}, −a_{2} + a_{5}, −a_{7} + a_{8}, a_{2} + a_{7} − a_{8}, −a_{3} + a_{6}, a_{7}, a_{10}, a_{3} − a_{7} − a_{10}, a_{9}).

It can be checked that if p is taken as (a_{1}, a_{2}, a_{3}, a_{7}, 0, a_{7}, a_{8}−a_{7}, 0, a_{10}, 0) then p satisfies the inequalities as given above and f = c. Furthermore, xT x^{t} = 0, so that λ_{p,a} ∈ 1 + vℤ[v]. Since for any p satisfying the above inequalities we must have λ_{a,p} ∈ ℤ[v] and the coefficient of the lowest power of v in λ_{a,p} is always 1, it follows that the coefficient of F_{l}^{c} in the above sum, when terms in the same PBW basis element are collected together, is of the form e + vh, where h ∈ ℤ[v] and e ∈ ℕ, e ≠ 0. Since b ∈ B, we know by [8, §§2.3, 3.2] that b has a unique expression as a ℤ[v]-combination of elements F_{l}^{d} as d varies, in which precisely one of the coefficients lies in 1 + vℤ[v]. It follows that e = 1 and b ≡ F_{l}^{c} mod vL, whence b ≡ F_{l}^{c} mod vL′ by [9, §2.3]. But we know that

F̃_{i_{1}}^{a_{1}} F̃_{i_{2}}^{a_{2}} ⋯ F̃_{i_{10}}^{a_{10}} ≡ F_{l}^{c} mod vL′,

since S_{i}^{l}(a) = c. It follows that

F̃_{i_{1}}^{a_{1}} F̃_{i_{2}}^{a_{2}} ⋯ F̃_{i_{10}}^{a_{10}} · 1 ≡ F_{i_{1}}^{(a_{1})}F_{i_{2}}^{(a_{2})} ⋯ F_{i_{10}}^{(a_{10})} mod vL′.□

As a consequence of Theorem 8.4 we may deduce the relationships between the set M_{i} of tight monomials, the Lusztig cone C_{i}, and the simplicial region of linearity X_{i} of the piecewise-linear function R_{j}^{j′}.

Theorem 8.5 For each reduced expression i for w_{0} let M_{i} = {F_{i_{1}}^{(a_{1})}F_{i_{2}}^{(a_{2})} ⋯ F_{i_{k}}^{(a_{k})} : a ∈ C_{i}}. Suppose we have type A_{4}. Then M_{i} ⊆ B. Moreover, under Kashiwara’s parametrization of B we have ψ_{i}(M_{i}) = C_{i} and under Lusztig’s parametrization of B we have ϕ_{j}(M_{i}) = X_{i}.

Proof: We recall that M_{i} ⊆ B was shown in [14]. Theorem 8.4 shows that ψ_{i}(M_{i}) = C_{i}. Proposition 6.1 shows that S_{i}^{j}(C_{i}) = X_{i}. Now S_{i}^{j} = ϕ_{j}ψ_{i}^{−1}. Since ψ_{i}^{−1}(C_{i}) = M_{i} it follows that ϕ_{j}(M_{i}) = X_{i}. □

Acknowledgements

The research for this paper was supported in part by EPSRC research grant ref. GR/K55233 (the second author was an EPSRC research assistant of Professor K. A. Brown at the University of Glasgow, Scotland). It was typeset in L^{A}TEX.

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