THE INTERSECTION OF OPPOSED BIG CELLS IN THE REAL FLAG VARIETY OF TYPE G_{2}

R. J. MARSH AND K. RIETSCH

Abstract. We compute the Euler characteristics of the individual connected components of the intersection of two opposed big cells in the real flag variety of type G_{2}, verifying a conjecture from [6].

1. Introduction

Let G be a connected linear algebraic group. Any flag variety (homogeneous projective variety G/B) has a myriad of cell decompositions, so–called ‘Bruhat decompositions’. For every Borel subgroup there is precisely one, and the cells in the decomposition are simply the orbits under this group. Furthermore there is always a unique open dense orbit called the ‘big cell’. In the present paper we fix two Borel subgroups opposite to one another and study the intersection of the two resulting big cells. More precisely we are interested in the real points of this variety, in the case where everything is split over ℝ.

We let B^{∗} denote the real points of the intersection of two opposed big cells, endowed with the Hausdorff topology coming from ℝ. There has been some recent work on determining the number of connected components and the Euler characteristics of these varieties B^{∗}, see e.g. [6, 5], and for the type A case [7, 8]. In particular it was open for a long time whether the connected components are always contractible. This has proved not to be the case. The first example is in type G_{2} found in [6] (but it is also false in type A for rank > 4 as follows from the explicit formula for the number of connected components in [8] and a computation of the Euler characteristics e.g. using [6]).

Therefore to the graph defined in [6] parameterizing the connected components of B^{∗} one can in principle add another nontrivial datum. That is, every component of the graph has an integer associated to it given simply by the Euler characteristic of the corresponding connected component of B^{∗}. Lusztig showed in [4] that the graphs arising in this setting are ‘mod 2’ quotients of the graphs parameterizing the canonical basis of the corresponding quantum enveloping algebra U^{−}. Therefore by taking the Euler characteristics one is assigning in a natural way an integer to every canonical basis element. No direct canonical basis interpretation of these integers is known. Moreover they have not up to now been computed in any nontrivial examples.

We focus our attention here on the G_{2} case. In Figure 2 we have reproduced the parameterization [6] of the connected components of B^{∗} in type G_{2}. There are 11 connected components, and one is conjectured to have Euler characteristic 2 (the last one), while the others should have Euler characteristic 1. By an independent method the total Euler characteristic of B^{∗} was worked out in [6] as 12.

The second named author was supported by EPSRC fellowship GR/M09506/01 and both authors were supported by a University of Leicester Research Fund Grant.

The aim of this paper is to compute the individual Euler characteristics of the connected components of B^{∗} to verify the above conjecture. We use two main tools. The first one is a cell decomposition of B^{∗} due to Deodhar [2], which is recalled in the next section. And the second ingredient is Berenstein and Zelevinsky’s Chamber Ansatz, explained in Section 3, which we require to be able to tell which connected component each of the various cells of Deodhar’s decomposition lie in. The resulting decompositions for the connected components are indicated in Figure 6. The Euler characteristics are the expected ones as can be read off from that figure.

2. Decomposition of B^{∗}

2.1. Preliminaries. We begin by introducing some notation and standard facts as can be found e.g. in [9]. Let G_{ℂ} be a complex simple linear algebraic group. After this general section we will choose it specifically as the one of type G_{2}. In any case we always focus on its split real form G_{ℝ} = G. All the varieties in this paper will be defined over ℝ, and we identify them with their ℝ–valued points. The topology we consider is the usual Hausdorff topology coming from ℝ (rather than Zariski topology). We write ℝ^{∗} for ℝ \ {0}.

Let B denote the (real) flag variety of G. The elements of B are the Borel subgroups of G. We sometimes write [B] for the Borel subgroup B considered as a point in the flag variety. The transitive action of G on B is denoted by g · [B] := [gBg^{−1}].

Let T ⊂ G be a fixed split maximal torus, and B^{+} a fixed Borel subgroup which contains T. We also automatically have given the opposite Borel subgroup B^{−} (such that B^{+} ∩ B^{−} = T) and the unipotent radicals U^{+} and U^{−} of these two Borel subgroups.

Let X^{∗}(T) and X_{∗}(T) be the character, respectively cocharacter lattices of T with their canonical perfect pairing

⟨ , ⟩ : X_{∗}(T) × X^{∗}(T) → ℤ.

And let A = (A_{ij}) be the Cartan matrix of G. We denote the positive simple roots (corresponding to B^{+}) by α_{1}, …, α_{r} ∈ X^{∗}(T). We will write α > 0 if α ∈ X^{∗}(T) is a positive root, i.e. a nonnegative linear combination of the simple roots. We also have the simple coroots α_{1}^{∨}, …, α_{r}^{∨} ∈ X_{∗}(T) which are determined by ⟨α_{i}^{∨}, α_{j}⟩ = A_{ij}.

We will make extensive use of the 1–parameter simple root subgroups: Let us fix for i = 1, …, r, Chevalley generators e_{i} and f_{i} ∈ Lie(G) of the (real) Lie algebra, with the e_{i}’s lying in positive simple root spaces. Then we define

x_{i} : ℝ → B^{+},  x_{i}(t) := exp(te_{i}),
y_{i} : ℝ → B^{−},  y_{i}(t) := exp(tf_{i}).

INTERSECTION OF OPPOSED REAL BIG CELLS FOR G_{2} 3

The Weyl group N_{G}(T)/T of G is denoted W and is generated by the usual simple reflections s_{1}, …, s_{r} (r being the rank of G). We fix a representative ṡ_{i} ∈ G for the Weyl group element s_{i} by defining ṡ_{i} := x_{i}(1)y_{i}(−1)x_{i}(1). For any other w ∈ W choose a reduced (minimal number of factors) expression w = s_{i_{1}} … s_{i_{k}} and set ẇ := ṡ_{i_{1}} … ṡ_{i_{k}} to get a representative. It is well known that this definition of ẇ is independent of the reduced expression. Note that x_{i}(t) ∈ U^{+} ∩ B^{−}ṡ_{i}B^{−} and y_{i}(t) ∈ U^{−} ∩ B^{+}ṡ_{i}B^{+}, whenever t ≠ 0. Explicitly,

x_{i}(t) = y_{i}(t^{−1}) ṡ_{i}α_{i}^{∨}(−t) y_{i}(t^{−1}),
y_{i}(t) = x_{i}(t^{−1}) α_{i}^{∨}(−t)ṡ_{i} x_{i}(t^{−1}).

The Weyl group acts on T and hence naturally also on X_{∗}(T) and X^{∗}(T). The length of (a reduced expression of) an element w ∈ W is denoted ℓ(w). We also consider the Bruhat order < on W. The statement ws_{i} < w is equivalent to w · α_{i} < 0, that w sends α_{i} to a negative root.

There is a unique longest element of W which is called w_{0}. We propose to study the open subvariety of the flag variety

B^{∗} := B^{+}ẇ_{0} · [B^{+}] ∩ B^{−}ẇ_{0} · [B^{−}] ⊂ B,

in other words the intersection of the two big cells for the Bruhat decompositions relative to B^{−}, respectively B^{+}.

Note that B^{∗} is an affine variety and can be identified naturally with open subvarieties both of U^{+} and of U^{−}. We have two isomorphisms

\begin{aligned}& & B^{∗} \\ & i_{+} ↗ & & ↖ i_{−} \\ U^{+} ∩ (B^{−}ẇ_{0}B^{−}) & & & & U^{−} ∩ (B^{+}ẇ_{0}B^{+}), \end{aligned} (2.1)

where i_{+} takes u ∈ U^{+} to u · [B^{−}] while the right hand map i_{−} takes u ∈ U^{−} to u · [B^{+}].

2.2. Deodhar’s decomposition of B^{∗}. We begin by making the following definition (see [3, Appendix]).

Definition 2.1. (relative position in B) Let w ∈ W and consider the action of G on B × B by simultaneous conjugation. We say that two Borel subgroups B, B′ ∈ B are in relative position w if the pair (B, B′) lies in the G–orbit of (B^{+}, ẇ · B^{+}). From Bruhat decomposition it follows that such w is unique and exists for any pair

(B, B′). We write B ⟶^{w} B′.

If B ⟶^{w} B′ and w = s_{i_{1}} … s_{i_{n}} is a reduced expression, then it also follows from Bruhat decomposition that there exist uniquely determined Borel subgroups B_{0}, B_{1}, …, B_{n} = B′ such that

B = B_{0} ⟶^{s_{i_{1}}} B_{1} ⟶^{s_{i_{2}}} B_{2} ⟶ ⋯ ⟶ B_{n−1} ⟶^{s_{in}} B_{n} = B′

Explicitly, B ⟶^{w} B′ means B = g · B^{+} and B′ = gẇ · B^{+} for some g ∈ G. And the B_{j}’s are then given by B_{j} = gṡ_{i_{1}} … ṡ_{i_{j}} · B^{+}.

Using the above notation, we have B^{∗} = {B ∈ B | B^{+} ⟶^{w_{0}} B ⟵^{w_{0}} B^{−}}. The following definition is a special case of [2, Definition 2.3].

Definition 2.2 (distinguished subexpressions for 1). Let s_{i_{1}} … s_{i_{n}} = w ∈ W be a reduced expression for w (so n = ℓ(w)). Then by a subexpression of this reduced expression we mean a sequence of Weyl group elements σ = (σ_{0}, σ_{1}, …, σ_{n}) such that

σ_{0} = 1,  σ_{j} = \begin{cases}either & σ_{j−1}s_{i_{j}} \\ or & σ_{j−1}, \end{cases}  for all j = 1, … n.

In particular the “empty” subexpression σ = (1, …, 1) is allowed. We call σ a subexpression for 1 if σ_{n} = 1.

A subexpression is called distinguished if we have

σ_{j} ≤ σ_{j−1} s_{i_{j}}, for all j ∈ {1, …, n}.

This means that if right multiplication by s_{i_{j}} decreases the length of σ_{j−1}, then we must choose σ_{j} = σ_{j−1}s_{i_{j}} to get a distinguished subexpression.

The following result is stated by Deodhar over an algebraically closed field, but it extends trivially to any split form, so we state it here in our present setting over the reals.

Theorem 2.3 ([2]). Let s_{i_{1}} … s_{i_{N}} be a fixed reduced expression for w_{0}.

1. Suppose B = xẇ_{0} ·B^{+} is an element of B^{∗}, where x ∈ B^{+}. Then the sequence (σ_{0}, …, σ_{N}) =: σ(B) defined by

xs_{i_{1}} … s_{i_{k}} ∈ B^{−}σ_{k}B^{+}

is a well–defined distinguished subexpression for 1 (of s_{i_{1}} … s_{i_{N}}).

2. Let D_{σ} := {B ∈ B^{∗} | σ(B) = σ}. Then

D_{σ} ≅ (ℝ^{∗})^{|I(σ)|} × ℝ^{|K(σ)|}, where I(σ) = {j ∈ {1, …, N} | σ_{j} = σ_{j−1}},
and K(σ) = {j ∈ {1, …, N} | σ_{j} < σ_{j−1}}.

An isomorphism as in part (2) of the theorem (albeit not identical to the one in [2]) will be constructed explicitly below. We continue to fix the reduced expression s_{i_{1}} … s_{i_{N}} for w_{0} in what follows.

Remark 2.4. The definition of D_{σ} is natural to state using relative position. If B ∈ B^{∗}, and B_{1}, …, B_{N} are defined by

B^{+} ⟶^{s_{i_{1}}} B_{1} ⟶^{s_{i_{2}}} B_{2} ⟶ ⋯ ⟶ B_{N−1} ⟶^{s_{i_{N}}} B_{N} = B,

then B ∈ D_{σ} precisely if B^{−} ⟶^{w_{0}σ_{j}} B_{j} for all j. So the σ controls the relative positions of the intermediate B_{i}’s with respect to B^{−}.

2.3. Inductive construction. Let σ be a fixed distinguished subexpression for 1 of s_{i_{1}} … s_{i_{N}}. We now want to describe D_{σ}^{−} ⊂ U^{−} ∩ B^{+}w_{0}B^{+}, the preimage of D_{σ} under the isomorphism i_{−} in (2.1). We will do this by building up the elements B ∈ D_{σ} from the B_{j}’s defined in the previous remark. To begin with note that

if B_{j−1} = g · B^{+}, then B_{j−1} ⟶^{s_{i_{j}}} B_{j} just says that B_{j} = gx_{i_{j}}(t)ṡ_{i_{j}} · B^{+} for some

t ∈ ℝ, or equivalently

B_{j} = \begin{cases}either & gy_{i_{j}}(t) · B^{+} some t ∈ ℝ^{∗}, \\ or & gṡ_{i_{j}} · B^{+}. \end{cases}

Let us first determine the possible B_{1}’s. There are two cases.

• If σ_{1} = 1, then we have B^{+} ⟶^{s_{i_{1}}} B_{1} ⟵^{w_{0}} B^{−}. Therefore B_{1} = y_{i_{1}}(t) · B^{+} for some t ∈ ℝ^{∗}.

• If on the other hand σ_{1} = s_{i_{1}} then B^{+} ⟶^{s_{i_{1}}} B_{1} ⟵^{w_{0}s_{i_{1}}} B^{−} and we get B_{1} = ṡ_{i_{1}} · B^{+}.

Suppose in general we have B_{j−1} = g · B^{+} given, where g = yσ̇_{j−1} for some y ∈ U^{−}. We then want to construct all possible B_{j} = g′ ·B^{+} from this B_{j−1}. There are three cases.

1. Suppose first that σ_{j} = σ_{j−1}. Then we have the setting

B^{−}  ⟶^{w_{0}σ_{j}} B_{j−1}
w_{0}σ_{j} ↘  ↙_{s_{i_{j}}}
B_{j}

So if B_{j−1} = g · B^{+} where g = yσ̇_{j−1}, then it is easy to rule out gṡ_{i_{j}} · B^{+} for B_{j} and the only possible solutions are of the form B_{j} = gy_{i_{j}}(t) · B^{+} for some t ∈ ℝ^{∗}. Note that w_{0}σ_{j}s_{i_{j}} < w_{0}σ_{j} (since σ is distinguished). We claim that therefore B_{j} = gy_{i_{j}}(t) · B^{+} has the correct relative positions for any t ∈ ℝ^{∗} and in fact g′ := gy_{i_{j}}(t) ∈ U^{−}σ̇_{j}. All of this follows since we have σ_{j−1} · α_{i_{j}} > 0 and therefore

gy_{i_{j}}(t) = yσ̇_{j−1}y_{i_{j}}(t) = yσ̇_{j−1}y_{i_{j}}(t)σ̇_{j−1}^{−1}σ̇_{j} ∈ U^{−}σ̇_{j},

using σ̇_{j−1} = σ̇_{j}.

2. Suppose next that σ_{j} > σ_{j−1}. Then

B^{−}  ⟶^{w_{0}σ_{j−1}} B_{j−1}
w_{0}σ_{j} ↘  ↗_{s_{i_{j}}}
B_{j}

and since the lengths add, ℓ(w_{0}σ_{j}) + ℓ(s_{i_{j}}) = ℓ(w_{0}σ_{j−1}), we get that B_{j} is uniquely determined by B_{j−1} and equals to gṡ_{i_{j}} · B^{+}. We immediately have that g′ := gṡ_{i_{j}} ∈ U^{−}σ̇_{j}.

3. Finally we have the case σ_{j} < σ_{j−1}. In this case the other two lengths add, ℓ(w_{0}σ_{j−1}) + ℓ(s_{i_{j}}) = ℓ(w_{0}σ_{j}), and the diagram

B^{−}  ⟶^{w_{0}σ_{j−1}} B_{j−1}
w_{0}σ_{j} ↘  ↙_{s_{i_{j}}}
B_{j}

is automatically satisfied for any B_{j} in position s_{i_{j}} relative to B_{j−1}. Therefore we can take B_{j} = gx_{i_{j}}(m)ṡ_{i_{j}} · B^{+} for any m ∈ ℝ.

We claim that in this case g′ := gx_{i_{j}}(m)ṡ_{i_{j}}^{−1} ∈ U^{−}σ̇_{j}. This holds because we have σ_{j−1} · α_{i_{j}} < 0. Therefore σ̇_{j−1}x_{i_{j}}(t)σ̇_{j−1}^{−1} ∈ U^{−} and

gx_{i_{j}}(m)ṡ_{i_{j}}^{−1} = yσ̇_{j−1}x_{i_{j}}(m)ṡ_{i_{j}}^{−1} = yσ̇_{j−1}x_{i_{j}}(m)σ̇_{j−1}^{−1}σ̇_{j} ∈ U^{−}σ̇_{j}.

Applying this procedure recursively to express finally B_{N} we get the following proposition (and we also recover Deodhar’s result Theorem 2.3).

Proposition 2.5. For w_{0} = s_{i_{1}} … s_{i_{N}} a fixed reduced expression, and σ a distinguished subexpression for 1 let

I(σ) = {j : 1 ≤ j ≤ N and σ_{j−1} = σ_{j}},
J(σ) = {j : 1 ≤ j ≤ N and σ_{j−1} < σ_{j}},
K(σ) = {j : 1 ≤ j ≤ N and σ_{j−1} > σ_{j}}.

Then we have explicitly

D_{σ}^{−} = \{z_{1}z_{2} … z_{N} | z_{j} = \begin{cases}y_{i_{j}}(t_{j}) & if j ∈ I(σ) \\ ṡ_{i_{j}} & if j ∈ J(σ) \\ x_{i_{j}}(m_{j})ṡ_{i_{j}}^{−1} & if j ∈ K(σ) \end{cases} , where m_{j} ∈ ℝ, t_{j} ∈ ℝ^{∗} \} .

Proof. By the inductive construction in 2.3 above we showed that any B ∈ D_{σ} is of the form z ·B^{+} for z = z_{1} … z_{N} as in the proposition (and that these z ·B’s all lie in D_{σ}). We also showed that z_{1} … z_{k} ∈ U^{−}σ̇_{k} for all k, so in particular z_{1} … z_{N} ∈ U^{−}, since σ_{N} = 1. Therefore D_{σ}^{−} is precisely the set of these z = z_{1} … z_{N}.

Remark 2.6. Note that the map

(ℝ^{∗})^{I(σ)} × ℝ^{K(σ)} → D_{σ}^{−}

arising from the proposition is an isomorphism, with inverse basically constructed in 2.3.

2.4. Further refinement. Over ℝ it is natural to consider the connected components of the D_{σ}’s to get a cell decomposition. Let h : I(σ) → {1, −1} be a choice of signs for the elements of I(σ). Then we define

D_{σ}^{−}(h) = \{z_{1}z_{2} … z_{N} | z_{j} = \begin{cases}y_{i_{j}}(t_{j}) with h(j)t_{j} ∈ ℝ_{>0} & if j ∈ I(σ), \\ ṡ_{i_{j}} & if j ∈ J(σ), \\ x_{i_{j}}(m_{j})ṡ_{i_{j}}^{−1} with m_{j} ∈ ℝ & if j ∈ K(σ). \end{cases} \} .

D_{σ}^{−}(h) is a (real) semi–algebraic cell in U^{−} ∩B^{+}ẇ_{0}B^{+} of dimension |I(σ)|+|K(σ)|. Its image i_{−}(D_{σ}^{−}(h)) in B^{∗} is denoted D_{σ}(h).

3. Type G_{2}

From now on let G be of type G_{2}. Then the Cartan matrix A = (A_{ij}) for type G_{2} is given by

A = (\begin{matrix}2 & −3 \\ −1 & 2 \end{matrix}) .

The Weyl group W has two generators s_{1}, s_{2} corresponding to reflection by the short root α_{1} and the long root α_{2}, respectively (see Figure 3). The fundamental weights are ω_{1} = ε_{1}, giving rise to a 7–dimensional representation, and ω_{2} = 2ε_{1} + ε_{2}, the highest weight of the 14–dimensional adjoint representation.

The longest element is

w_{0} = s_{1}s_{2}s_{1}s_{2}s_{1}s_{2} = s_{2}s_{1}s_{2}s_{1}s_{2}s_{1}.

We let i = (1, 2, 1, 2, 1, 2) and ĩ = (2, 1, 2, 1, 2, 1) stand for these two reduced expressions of w_{0}.

To give Deodhar’s decomposition of B^{∗} in this case we first need to fix a reduced expression of w_{0}, so let us pick i = (1, 2, 1, 2, 1, 2). Next we list all of the distinguished subexpressions for 1 of i:

Σ_{i} := \{ \begin{matrix}xxxxxx := (1, 1, 1, 1, 1, 1, 1), & xxx2x2 := (1, 1, 1, 1, s_{2}, s_{2}, 1), \\ 1x1xxx := (1, s_{1}, s_{1}, 1, 1, 1, 1), & 1x12x2 := (1, s_{1}, s_{1}, 1, s_{2}, s_{2}, 1), \\ x2x2xx := (1, 1, s_{2}, s_{2}, 1, 1, 1), & 12x21x := (1, s_{1}, s_{1}s_{2}, s_{1}s_{2}, s_{1}, 1, 1), \\ xx1x1x := (1, 1, 1, s_{1}, s_{1}, 1, 1), & x21x12 := (1, 1, s_{2}, s_{2}s_{1}, s_{2}s_{1}, s_{2}, 1) \end{matrix} \} .

INTERSECTION OF OPPOSED REAL BIG CELLS FOR G_{2} 7

Figure 1. G_{2} root system

Furthermore, define

y_{xxxxxx}(t_{1}, …, t_{6}) := y_{i}(t_{1}, …, t_{6}) := y_{1}(t_{1})y_{2}(t_{2})y_{1}(t_{3})y_{2}(t_{4})y_{1}(t_{5})y_{2}(t_{6})
y_{1x1xxx}(t_{1}, m_{1}, t_{2}, t_{3}, t_{4}) := ṡ_{1}y_{2}(t_{1})x_{1}(m_{1})ṡ_{1}^{−1}y_{2}(t_{2})y_{1}(t_{3})y_{2}(t_{4})
y_{x2x2xx}(t_{1}, t_{2}, m_{1}, t_{3}, t_{4}) := y_{1}(t_{1})ṡ_{2}y_{1}(t_{2})x_{2}(m_{1})ṡ_{2}^{−1}y_{1}(t_{3})y_{2}(t_{4})
y_{xx1x1x}(t_{1}, t_{2}, t_{3}, m_{1}, t_{4}) := y_{1}(t_{1})y_{2}(t_{2})ṡ_{1}y_{2}(t_{3})x_{1}(m_{1})ṡ_{1}^{−1}y_{2}(t_{4})
y_{xxx2x2}(t_{1}, t_{2}, t_{3}, t_{4}, m_{1}) := y_{1}(t)y_{2}(t_{2})y_{1}(t_{3})ṡ_{2}y_{1}(t_{4})x_{2}(m_{1})ṡ_{2}^{−1}
y_{1x12x2}(t_{1}, m_{1}, t_{2}, m_{2}) := ṡ_{1}y_{2}(t_{1})x_{1}(m_{1})ṡ_{1}^{−1}ṡ_{2}y_{1}(t_{2})x_{2}(m_{2})ṡ_{2}^{−1}
y_{12x21x}(t_{1}, m_{1}, m_{2}, t_{2}) := ṡ_{1}ṡ_{2}y_{1}(t_{1})x_{2}(m_{1})ṡ_{2}^{−1}x_{1}(m_{2})ṡ_{1}^{−1}y_{2}(t_{2})
y_{x21x12}(t_{1}, t_{2}, m_{1}, m_{2}) := y_{1}(t_{1})ṡ_{2}ṡ_{1}y_{2}(t_{2})x_{1}(m_{1})ṡ_{1}^{−1}x_{2}(m_{2})ṡ_{2}^{−1}.

By Proposition 2.5 we have

D_{i}^{−} := D_{xxxxxx}^{−} = {y_{i}(t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}) | t_{1}, …, t_{6} ∈ ℝ^{∗}}
D_{1x1xxx}^{−} = {y_{1x1xxx}(t_{1}, m_{1}, t_{2}, t_{3}, t_{4}) | t_{1}, …, t_{4} ∈ ℝ^{∗}, m_{1} ∈ ℝ}
⋮
D_{x21x12}^{−} = {y_{x21x12}(t_{1}, t_{2}, m_{1}, m_{2}) | t_{1}, t_{2} ∈ ℝ^{∗}, m_{1}, m_{2} ∈ ℝ},

with the property that

U^{−} ∩ B^{+}ẇ_{0}B^{+} = ⨆_{σ∈Σ_{i}} D_{σ}^{−}.

From these we also get the D_{σ}^{−}(h)’s and D_{σ}(h)’s defined in 2.4, where h : I(σ) → {1, −1} determines the signs of the t_{i} ∈ ℝ^{∗}.

4. Parameterization of the connected components of B^{∗} : using the opposite reduced expression

We recall the parameterization of the set of connected components of B^{∗} from [6]. Consider the open subset D_{i} = ⨆_{h:{1,...,6}→{±1}} D_{i}(h) of B^{∗} from above. Explicitly,

D_{i}(h) = {y_{i}(t_{1}, …, t_{6}) · [B^{+}] = y_{1}(t_{1})y_{2}(t_{2}) … y_{1}(t_{5})y_{2}(t_{6}) · [B^{+}] | h(i)t_{i} ∈ ℝ_{>0}} .

We also have analogs of these for ĩ = (2, 1, 2, 1, 2, 1),

Dĩ(h) := {yĩ(t_{1}, …, t_{6}) · [B^{+}] := y_{2}(t_{1}) … y_{1}(t_{6}) · [B^{+}] | h(i)t_{i} ∈ ℝ_{>0}}
Dĩ := ⨆_{h:{1,...,6}→{±1}} Dĩ(h).

It was proved in [6] that the union D^{∗} := D_{i} ∪ Dĩ has complement of codimension ≥ 2 in B^{∗}. Hence the connected components of B^{∗} correspond bijectively to the connected components of D^{∗}. And these were determined roughly by checking which of the Dĩ(h′)’s overlap with which D_{i}(h)’s. Figure 2 adapted from [6] shows which D_{i}(h)’s and Dĩ(h′)’s lie in the same connected component, and these components are numbered for later use. The sequences of signs in the figure stand for the values of h : {1, …, 6} → {±1}, and the columns indicate whether D_{i}(h) or Dĩ(h) is meant. So for example the sequence of signs in the third row first column stands for D_{i}((1, 1, −1, 1, −1, 1)), and the figure says that this cell lies in connected component 5.

5. Berenstein–Zelevinsky’s generalized chamber Ansatz for G_{2}

We wish to determine which connected component of B^{∗} each D_{σ}(h) lies in. We shall do this by proving that each D_{σ}(h) actually intersects one of the cells Dĩ(h′) (for some choice h′ of signs) listed in the previous section. It then follows that D_{σ}(h) lies in the same connected component as this cell.

In order to show that an element y ∈ D_{σ}^{−} also lies in one of the cells D− ; ĩ(h′), we need to show that y can be expressed in the form y = yĩ(a_{1}, …, a_{6}) = y_{2}(a_{1})y_{1}(a_{2})y_{2}(a_{3})y_{1}(a_{4})y_{2}(a_{5})y_{1}(a_{6}) for some non–zero real numbers a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6} ∈ ℝ^{∗} with signs h′(1), h′(2), h′(3), h′(4), h′(5), h′(6). We can then deduce which connected component D_{σ}(h) lies in.

There already exists in the literature a beautiful method for determining a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, should they exist — this is the Chamber Ansatz of Berenstein, Fomin and Zelevinsky; we therefore employ this method.

There is a natural map ε : U^{+} ∩ B^{−}ẇ_{0}B^{−} → U^{−} ∩ B^{+}ẇ_{0}B^{+}, given explicitly by ε = i_{−}^{−1} ∘ i_{+} (see 2.1); similarly we define α : U^{−} ∩ B^{+}wB^{+} → U^{+} ∩ B^{−}w^{−1}B^{−} to be α = i_{+}^{−1} ∘ i_{−}, the inverse of ε.

U^{+} ∩ B^{−}ẇ_{0}B^{−} ⟶_{ε} U^{−} ∩ B^{+}ẇ_{0}B^{+}

U^{+} ∩ B^{−}ẇ_{0}B^{−} ⟵_{α} U^{−} ∩ B^{+}ẇ_{0}B^{+}

\begin{aligned}& D_{i}(h)  Dĩ(h) & D_{i}(h)  Dĩ(h) \\ 1 & + + + + + + + + + + ++ & − − − − − − − − − − −− & 2 \end{aligned}
3  + − + − + − − + − + −+  − + − + − + + − + − +−  4

5  \begin{matrix}+ + − + −+ & + − + − ++ \\ − − + + −+ & + − + + −− \\ − + − − −+ & + − − − +− \\ − − − − ++ & + + − − −− \\ − + − + +− & − + + − +− \\ − − − + −− & − − + − −− \\ − + + − −− & − − − + +− \\ + − − − −− & − − − − −+ \end{matrix}  \begin{matrix}− − + − +− & − + − + −− \\ + + − − +− & − + − − ++ \\ + − + + +− & − + + + −+ \\ + + + + −− & − − + + ++ \\ + − + − −+ & + − − + −+ \\ + + + − ++ & + + − + ++ \\ + − − + ++ & + + + − −+ \\ − + + + ++ & + + + + +− \end{matrix}  6

7  \begin{matrix}+ − + − ++ & + + − + −+ \\ + − + + −− & − − + + −+ \\ + − − − +− & − + − − −+ \\ + + − − −− & − − − − ++ \\ − + + − +− & − + − + +− \\ − − + − −− & − − − + −− \\ − − − + +− & − + + − −− \\ − − − − −+ & + − − − −− \end{matrix}  \begin{matrix}− + − + −− & − − + − +− \\ − + − − ++ & + + − − +− \\ − + + + −+ & + − + + +− \\ − − + + ++ & + + + + −− \\ + − − + −+ & + − + − −+ \\ + + − + ++ & + + + − ++ \\ + + + − −+ & + − − + ++ \\ + + + + +− & − + + + ++ \end{matrix}  8

9  \begin{matrix}− + − + ++ & + + + + −+ \\ + − − + +− & + + − + −− \\ − + + + +− & + + − − ++ \\ − − − + −+ & + − + + ++ \end{matrix}  \begin{matrix}+ − + − −− & − − − − +− \\ − + + − −+ & − − + − ++ \\ + − − − −+ & − − + + −− \\ + + + − +− & − + − − −− \end{matrix}

10  \begin{matrix}+ + + + −+ & − + − + ++ \\ + + − + −− & + − − + +− \\ + + − − ++ & − + + + +− \\ + − + + ++ & − − − + −+ \end{matrix}  \begin{matrix}− − − − +− & + − + − −− \\ − − + − ++ & − + + − −+ \\ − − + + −− & + − − − −+ \\ − + − − −− & + + + − +− \end{matrix}

11  \begin{matrix}− − − + ++ & + + + − −− \\ − + + − ++ & + + − + +− \\ + − − − ++ & + + − − −+ \\ + − + + −+ & + − + + −+ \\ − − + − −+ & + − − + −− \\ + + − − −+ & + − − − ++ \end{matrix}  \begin{matrix}+ + + − −− & − − − + ++ \\ + − − + −− & − − + − −+ \\ − + + + −− & − − + + +− \\ − + − − +− & − + − − +− \\ + + − + +− & − + + − ++ \\ − − + + +− & − + + + −− \end{matrix}

Figure 2. Connected components of B^{∗} identified with U^{−} ∩ B^{+}ẇ_{0}B^{+}, following [6]

Let y ∈ U^{−} ∩ B^{+}ẇ_{0}B^{+}. Then Theorem 1.4 in [1] can be used to calculate α(y) ∈ U^{+} ∩ B^{−}ẇ_{0}B^{−} as an element of the form

x = xĩ(a, b, c, d, e, f) := x_{2}(a)x_{1}(b)x_{2}(c)x_{1}(d)x_{2}(e)x_{1}(f).

A second application of this Theorem can be used to calculate y = ε(x) ∈ U^{−} ∩ B^{+}ẇ_{0}B^{+} as an element of the form yĩ(a′, b′, c′, d′, e′, f′). If the minors in the Chamber Ansatz do not vanish, then we have found a′, b′, c′, d′, e′, f′ ∈ ℝ^{∗} such that y = yĩ(a′, b′, c′, d′, e′, f′) as required.

We shall therefore need a description of the Chamber Ansatz in type G_{2}. If w ∈ W and i = 1, 2, then Berenstein and Zelevinsky define wω_{i} to be a chamber weight of level i, and define a corresponding ‘generalised minor’ Δ^{wω_{i}}, which is a function on G. These minors reduce to usual minors of a matrix in type A. The generalised minors can be described (using section 6 of [1]) in the following way. Let g ∈ G, w ∈ W, and ω a dominant weight. Then there is a module V_{ω} for G of highest weight ω. Fix a vector v_{ω} ∈ V_{ω} of highest weight ω, fix a reduced expression w = s_{j_{1}}s_{j_{2}} ⋯ s_{j_{l}} for w, and for k = 1, …, l set b_{k} = ⟨α_{j_{k}}^{∨}, s_{j_{k−1}} ⋯ s_{j_{1}}ω⟩. Then

v_{wω} := f_{j_{l}}^{(b_{l})}f_{j_{l−1}}^{(b_{(l−1)})} ⋯ f_{j_{1}}^{(b_{1})}v_{ω}

is an extremal weight vector of weight wω. Then Δ^{wω}(g) is defined to be the coefficient of v_{ω} in g · v_{wω}. We have:

Theorem 5.1. (Berenstein and Zelevinsky) Let x ∈ U^{+} ∩ B^{−}ẇ_{0}B^{−}, and suppose that j = (j_{1}, …, j_{N}) is a reduced expression for w_{0}. Then

ε(x) = y = y_{j_{1}}(a_{1})y_{j_{2}}(a_{2}) ⋯ y_{j_{N}} (a_{N}),

where a_{1}, a_{2}, …, a_{N} are given by:

a_{N+1−k} = \frac{1}{Δ^{w_{k}ω_{j_{k}}} (x)Δ^{w_{k+1}ω_{j_{k}}} (x)} ∏_{j≠j_{k}} Δ^{w_{k}ω_{j}}(x)^{−A_{j,j_{k}}} ,

where w_{k} = s_{j_{N}} s_{j_{N−1}} ⋯ s_{j_{k}}.

If we are in the situation of the Theorem, we write ε^{j}(x) = (a_{1}, a_{2}, …, a_{N}). Again using [1, Theorem 1.4], we can describe α using the Chamber Ansatz.

Define Δ_{−}^{wω}(g) to be the coefficient of v_{−ω} in g · v_{wω}, where v_{wω} is the extremal weight vector of weight wω as defined above.

Theorem 5.2. (Berenstein and Zelevinsky) Let y ∈ U^{−} ∩ B^{+}ẇ_{0}B^{+}, and suppose that j = (j_{1}, …, j_{N}) is a reduced expression for w_{0}. Then

α(y) = x = x_{j_{1}}(a_{1})x_{j_{2}}(a_{2}) ⋯ x_{j_{N}} (a_{N}),

where a_{1}, a_{2}, …, a_{N} are given by:

a_{N+1−k} = \frac{1}{Δ_{−}^{−w_{k}ω_{j_{k}}} (y)Δ_{−}^{−w_{k+1}ω_{j_{k}}} (y)} ∏_{j≠j_{k}} Δ_{−}^{−w_{k}ω_{j}}(y)^{−A_{j,j_{k}}} ,

where w_{k} = s_{j_{N}} s_{j_{N−1}} ⋯ s_{j_{k}}.

If we are in the situation of this Theorem, we write α^{j}(x) = (a_{1}, a_{2}, …, a_{N}). We can use a wiring diagram to describe these maps in type G_{2}, much as in

type A. Recall that our two reduced expressions of w_{0} are i = (1, 2, 1, 2, 1, 2) and ĩ = (2, 1, 2, 1, 2, 1). We begin by calculating need ε^{212121} = εĩ; this can be done using the Chamber Ansatz in Figure 5.

Suppose that a_{j} is one of the parameters in Theorem 5.1, and suppose that adjacent to the jth crossing from the left in Figure 5, chamber A is above, D below, and B and C on the same horizontal level. Let Δ(A), Δ(B), Δ(C), Δ(D)

Figure 3. The Chamber Ansatz for ε^{212121}.

Figure 4. The Chamber Ansatz for α^{212121}.

be the generalized minors corresponding to these four chambers (taking the value 1 on unbounded chambers). Then Theorem 5.1 states that:

a_{j} = \frac{Δ(A)^{3}Δ(D)^{3}}{Δ(B)Δ(C)} .

We also need to calculate αĩ = α^{212121}, which can be done using the Chamber Ansatz in Figure 5. The value of the component at a crossing is calculated as for ε, but using this second picture.

6. The map i_{+}

Note that our approach to parameterizing B^{∗} in Section 4 was inherently asymmetric, suited to the map i_{−} : U^{−} ∩ B^{+}ẇ_{0}B^{+} → B^{∗}. That is, the cells D_{j}(h) came naturally from cells D_{j}^{−}(h) defined in U^{−}. We now need to consider another set of cells in B^{∗} coming from the isomorphism i^{+} : U^{+} ∩ B^{−}ẇ_{0}B^{−} → B^{∗}. Let us define

xĩ(a_{1}, …, a_{6}) := x_{2}(a_{1})x_{1}(a_{2})x_{2}(a_{3})x_{1}(a_{4})x_{2}(a_{5})x_{1}(a_{6}),
D+ ; ĩ(h) := {xĩ(a_{1}, …, a_{6}) | a_{i}h(i) ∈ ℝ_{>0}}.

for any map h : {1, 2, 3, 4, 5, 6} → {±1}. It is clear that two cells D+ ; ĩ(h) and D+ ; ĩ(h′) lie in the same connected component of U^{+} ∩ B^{+}ẇ_{0}B^{+} precisely if D− ; ĩ(h) and D− ; ĩ(h′) lie in the same component, by symmetry between U^{+} and U^{−}. Reading off from Figure 2 when this happens, we obtain the new Figure 5, which groups together all sequences of signs (determining maps h : {1, 2, 3, 4, 5, 6} → {±1}) such that the corresponding D+ ; ĩ(h)’s lie in the same connected component of U^{+} ∩B^{−}ẇ_{0}B^{−}. We have labeled these connected components by letters A–K.

Our aim is now to identify which connected components of U^{+}∩B^{−}ẇ_{0}B^{−} labeled by A–K correspond to which components labeled 1–11 of B^{∗} under the map i_{+}. To do this we simply need to evaluate the map ε^{212121} on 11 test points, one from each component of U^{+} ∩ B^{−}ẇ_{0}B^{−}.

It suffices to calculate ε^{212121} on elements x of the form x = xĩ(a, b, c, d, e, f). From the Chamber Ansatz we have

ε(xĩ(a, b, c, d, e, f)) = yĩ(a′, b′, c′, d′, e′, f′),

A + + + + ++ − − − − −− B C − + − + −+ + − + − +− D

E \begin{matrix}+ − + − ++ \\ + − + + −− \\ + − − − +− \\ + + − − −− \\ − + + − +− \\ − − + − −− \\ − − − + +− \\ − − − − −+ \end{matrix} \begin{matrix}− + − + −− \\ − + − − ++ \\ − + + + −+ \\ − − + + ++ \\ + − − + −+ \\ + + − + ++ \\ + + + − −+ \\ + + + + +− \end{matrix} F G \begin{matrix}+ + − + −+ \\ − − + + −+ \\ − + − − −+ \\ − − − − ++ \\ − + − + +− \\ − − − + −− \\ − + + − −− \\ + − − − −− \end{matrix} \begin{matrix}− − + − +− \\ + + − − +− \\ + − + + +− \\ + + + + −− \\ + − + − −+ \\ + + + − ++ \\ + − − + ++ \\ − + + + ++ \end{matrix} H

I \begin{matrix}+ + + + −+ & − − − − +− \\ + + − + −− & − − + − ++ \\ + + − − ++ & − − + + −− \\ + − + + ++ & − + − − −− \end{matrix}  J \begin{matrix}− + − + ++ & + − + − −− \\ + − − + +− & − + + − −+ \\ − + + + +− & + − − − −+ \\ − − − + −+ & + + + − +− \end{matrix}

K \begin{matrix}+ + + − −− & − − − + ++ \\ + + − + +− & − − + − −+ \\ + + − − −+ & − − + + +− \\ + − + + −+ & − + − − +− \\ + − − + −− & − + + − ++ \\ + − − − ++ & − + + + −− \end{matrix}

Figure 5. The connected components of U^{+}∩B^{−}ẇ_{0}B^{−} in terms
of the cells D+ ; ĩ(h).

where

a′ = \frac{1}{Δ^{ε_{1}−ε_{3}}(x)Δ^{ε_{1}−ε_{2}}(x)},
b′ = \frac{Δ^{ε_{1}−ε_{2}}(x)}{Δ^{ε_{1}}(x)Δ^{−ε_{2}}(x)},
c′ = \frac{Δ^{−ε_{2}}(x)^{3}}{Δ^{ε_{1}−ε_{2}}(x)Δ^{ε_{3}−ε_{2}(x)}} ,
d′ = \frac{Δ^{ε_{3}−ε_{2}}(x)}{Δ^{−ε_{2}}(x)Δ^{ε_{3}}(x)},
e′ = \frac{Δ^{ε_{3}}(x)^{3}}{Δ^{ε_{3}−ε_{2}}(x)Δ^{ε_{3}−ε_{1}}(x)},
f′ = \frac{Δ^{ε_{3}−ε_{1}}(x)}{Δ^{ε_{3}}(x)Δ^{−ε_{1}}(x)}.

By considering the usual action of the Chevalley generators of the Lie algebra in the two fundamental representations V_{ω_{1}} and V_{ω_{2}} of G_{2}, the action of x_{1}(t),x_{2}(t), y_{1}(t),y_{2}(t),ṡ_{1} and ṡ_{2} can be computed explicitly. We list below the relevant matrix coefficients for the element x = xĩ(a, b, c, d, e, f).

Δ^{ε_{1}}(x) = 1,
Δ^{−ε_{3}}(x) = f + d + b,
Δ^{−ε_{2}}(x) = ed + eb + bc,
Δ^{ε_{2}}(x) = f^{2}ed + f^{2}eb + f^{2}bc + 2 bcdf + bcd^{2},
Δ^{ε_{3}}(x) = bcd^{2}e,
Δ^{−ε_{1}}(x) = bcd^{2}ef,
Δ^{ε_{1}−ε_{3}}(x) = 1,
Δ^{ε_{1}−ε_{2}}(x) = e + c + a,
Δ^{ε_{2}−ε_{3}}(x) = f^{3}e + f^{3}c + f^{3}a + 3 f^{2}cd + 3 f^{2}ad + 3 f^{2}ab + 3 fd^{2}c + 3 fd^{2}a +
+ 6fabd + 3 fab^{2} + d^{3}c + d^{3}a + 3 abd^{2} + 3 ab^{2}d + ab^{3},
Δ^{ε_{3}−ε_{2}}(x) = e^{2}d^{3}c + e^{2}d^{3}a + 3 e^{2}abd^{2} + 3 e^{2}ab^{2}d + e^{2}ab^{3} + 3 eab^{2}cd + 2 eab^{3}c
+ ab^{3}c^{2},
Δ^{ε_{2}−ε_{1}}(x) = f^{3}e^{2}d^{3}c + f^{3}e^{2}d^{3}a + 3 f^{3}e^{2}abd^{2} + 3 f^{3}e^{2}ab^{2}d + f^{3}e^{2}ab^{3} +
+ 3 f^{3}eab^{2}cd + 2 f^{3}eab^{3}c + f^{3}ab^{3}c^{2} + 3 f^{2}eab^{2}cd^{2} + 3 f^{2}eab^{3}cd+
+ 3 f^{2}ab^{3}c^{2}d + 3 ab^{3}c^{2}d^{2}f + ab^{3}c^{2}d^{3},
Δ^{ε_{3}−ε_{1}}(x) = ab^{3}c^{2}d^{3}e.

Applying the Chamber Ansatz, we obtain the components a′, b′, c′, d′, e′, f′:

a′ = \frac{1}{e + c + a},
b′ = \frac{e + c + a}{ed + eb + bc},
c′ = \frac{(ed + eb + bc)^{3}}{u (e + c + a)} ,
d′ = \frac{u}{bcd^{2}e (ed + eb + bc)},
e′ = \frac{e^{2}d^{3}c}{au},
f′ = \frac{ab}{def} ,

where

u = (e^{2}d^{3}c + e^{2}d^{3}a + 3 e^{2}abd^{2} + 3 e^{2}ab^{2}d + e^{2}ab^{3} + 3 eab^{2}cd + 2 eab^{3}c + ab^{3}c^{2}).

The next step is to substitute values for a, b, c, d, e, f with specific combinations of signs, one choice from each component of Figure 5, ensuring that the matrix coefficients in the Chamber Ansatz do not vanish. It turns out that this can be achieved by setting a = ±1, b = ±2, c = ±3, d = ±5, e = ±7 and f = ±11. For any choice of signs of a, b, c, d, e, f, we obtain the signs of a′, b′, c′, d′, e′, f′, and thus the connected component using Figure 2. The resulting bijection between connected components of U^{+} ∩ B^{−}ẇ_{0}B^{−} and B^{∗} is shown below.

\begin{matrix}A ⟷ 1 & B ⟷ 2 & C ⟷ 3 & D ⟷ 4 \\ E ⟷ 5 & F ⟷ 6 & G ⟷ 7 & H ⟷ 8 \end{matrix} ; I ⟷ 10 J ⟷ 9 ; K ⟷ 11 (6.1)

7. Calculation of the Euler Characteristics

We need to determine which connected components of U^{−} ∩B^{+}ẇ_{0}B^{+} the D_{σ}^{−}(h) belong to. If one point in such a D_{σ}^{−}(h) lies in a particular connected component, then the whole of it does. So a similar approach to that in the previous section will work. We start with a general point in D_{σ}^{−}(h) ⊆ U^{−} ∩ B^{+}ẇ_{0}B^{+}, apply α to it, to get a point in U^{+} ∩B^{−}ẇ_{0}B^{−}; we ensure this is of the form x_{212121}(a′, b′, c′, d′, e′, f′) for some a′, b′, c′, d′, e′, f′; then it lies in one of the components A–K of Figure 5; the bijection (6.1) above will then give us the connected component of D_{σ}^{−}(h). Again, not every point in D_{σ}^{−} need be in the domain of α^{212121} but we can always find one which is. We use the Chamber Ansatz for α^{212121} as described above. Our computation is as follows:

The following are the components of α^{212121} applied to y_{x21x12}(t_{1}, t_{2}, m_{1}, m_{2}) (provided this is well-defined).

a = −m_{2}^{−1},
b = \frac{m_{2}}{t_{1} m_{2} + m_{1}} ,
c = \begin{matrix}(t_{1} m_{2} + m_{1} ) & 3 \\ m_{2} & (−m_{2} t_{2} + m_{1}^{3}) \end{matrix},
d = \frac{−m_{2} t_{2} + m_{1}^{3}}{(t_{1} m_{2} + m_{1} ) (t_{1} m_{1}^{2} + t_{2} )},

e = −\begin{matrix}(t_{1} m_{1}^{2} + t_{2} ) & 3 \\ (−m_{2} t_{2} + m_{1}^{3}) & t_{2} & 2 \end{matrix} ,
f = \frac{t_{2}}{(t_{1} m_{1}^{2} + t_{2} ) t_{1}} .

The following are the components of α^{212121} applied to y_{12x21x}(t_{1}, m_{1}, m_{2}, t_{2}).

a = t_{2}^{−1},
b = −m_{2}^{−1},
c = \frac{m_{2}^{3}}{3 t_{1} m_{2} + m_{1}} ,
d = \frac{3 t_{1} m_{2} + m_{1}}{m_{2} (2 t_{1} m_{2} + m_{1} )},
e = (2 t_{1} m_{2} + m_{1} )^{3} (3 t_{1} m_{2} + m_{1} ) t_{1}^{3} ,
f = −\frac{t_{1}}{2 t_{1} m_{2} + m_{1}} .

The following are the components of α^{212121} applied to y_{1x12x2}(t_{1}, m_{1}, t_{2}, m_{2}).

a = −m_{2}^{−1},
b = −\frac{m_{2}}{m_{1} m_{2} + t_{2}} ,
c = −\begin{matrix}(m_{1} m_{2} + t_{2} ) & 3 \\ m_{2} & (t_{1} m_{2}^{2} − t_{2}^{3}) \end{matrix},
d = \frac{t_{1} m_{2}^{2} − t_{2}^{3}}{(m_{1} m_{2} + t_{2} ) (t_{1} m_{2} + m_{1} t_{2}^{2})},

e = (t_{1} m_{2} + m_{1} t_{2}^{2})^{3} ; (t_{1} m_{2}^{2} − t_{2}^{3}) t_{1} t_{2}^{3} ,
f = \frac{t_{2}^{2}}{t_{1} m_{2} + m_{1} t_{2}^{2}} .

The following are the components of α^{212121} applied to y_{xx1x1x}(t_{1}, t_{2}, t_{3}, m_{1}, t_{4}).

a = (t_{2} + t_{4} )^{−1} ,
b = \frac{t_{2} + t_{4}}{t_{1} t_{2} − m_{1} t_{4} + t_{4} t_{1}} ,

c = −  (t_{1} t_{2} − m_{1} t_{4} + t_{4} t_{1} )^{3} ; (t_{2} + t_{4} ) t_{4} (−t_{2} t_{3} + t_{4} m_{1}^{3}t_{2} − t_{4} t_{3} ),
d = −\frac{−t_{2} t_{3} + t_{4} m_{1}^{3}t_{2} − t_{4} t_{3}}{(t_{1} t_{2} m_{1}^{2} − t_{3} ) (t_{1} t_{2} − m_{1} t_{4} + t_{4} t_{1} )},

e = (t_{1} t_{2} m_{1}^{2} − t_{3} )^{3} t_{4} ; (−t_{2} t_{3} + t_{4} m_{1}^{3}t_{2} − t_{4} t_{3} ) t_{2} t_{3}^{2} ,
f = −\frac{t_{3}}{(t_{1} t_{2} m_{1}^{2} − t_{3} ) t_{1}} .

The following are the components of α^{212121} applied to y_{1x1xxx}(t_{1}, m_{1}, t_{2}, t_{3}, t_{4}).

a = (t_{2} + t_{4} )^{−1} ,
b = −\frac{t_{2} + t_{4}}{m_{1} t_{2} + m_{1} t_{4} − t_{4} t_{3}} ,

c = −  (m_{1} t_{2} + m_{1} t_{4} − t_{4} t_{3} )^{3} ; (t_{2} + t_{4} ) (t_{1} t_{2}^{2} + 2 t_{1} t_{2} t_{4} + t_{4}^{2}t_{1} + t_{2} t_{3}^{3}t_{4}^{2}),
d = \frac{t_{1} t_{2}^{2} + 2 t_{1} t_{2} t_{4} + t_{4}^{2}t_{1} + t_{2} t_{3}^{3}t_{4}^{2}}{(m_{1} t_{2} + m_{1} t_{4} − t_{4} t_{3} ) (t_{1} t_{2} + t_{4} t_{1} + t_{2} t_{3}^{2}t_{4} m_{1} )},

e = −  (t_{1} t_{2} + t_{4} t_{1} + t_{2} t_{3}^{2}t_{4} m_{1} )^{3} ; (t_{1} t_{2}^{2} + 2 t_{1} t_{2} t_{4} + t_{4}^{2}t_{1} + t_{2} t_{3}^{3}t_{4}^{2}) t_{1} t_{2}^{2}t_{3}^{3}t_{4} ,
f = \frac{t_{2} t_{3}^{2}t_{4}}{t_{1} t_{2} + t_{4} t_{1} + t_{2} t_{3}^{2}t_{4} m_{1}} .

The following are the components of α^{212121} applied to y_{xxx2x2}(t_{1}, t_{2}, t_{3}, t_{4}, m_{1}).

a = − (m_{1} − t_{2} )^{−1} ,
b = \frac{m_{1} − t_{2}}{t_{1} m_{1} + m_{1} t_{3} − t_{1} t_{2} − t_{4}} ,

c = (t_{1} m_{1} + m_{1} t_{3} − t_{1} t_{2} − t_{4} )^{3} ; (m_{1} − t_{2} ) (t_{2} t_{3}^{3}m_{1}^{2} − 3 t_{2} t_{3}^{2}t_{4} m_{1} + 3 t_{2} t_{3} t_{4}^{2} − t_{4}^{3}),
d = \frac{t_{2} t_{3}^{3}m_{1}^{2} − 3 t_{2} t_{3}^{2}t_{4} m_{1} + 3 t_{2} t_{3} t_{4}^{2} − t_{4}^{3}}{(t_{1} m_{1} + m_{1} t_{3} − t_{1} t_{2} − t_{4} ) (t_{1} t_{2} t_{3}^{2}m_{1} − 2 t_{1} t_{2} t_{3} t_{4} + t_{4}^{2}t_{1} + t_{3} t_{4}^{2})},

e = −  (t_{1} t_{2} t_{3}^{2}m_{1} − 2 t_{1} t_{2} t_{3} t_{4} + t_{4}^{2}t_{1} + t_{3} t_{4}^{2})^{3} ; (t_{2} t_{3}^{3}m_{1}^{2} − 3 t_{2} t_{3}^{2}t_{4} m_{1} + 3 t_{2} t_{3} t_{4}^{2} − t_{4}^{3}) t_{2} t_{3}^{3}t_{4}^{3} ,
f = t_{3} t_{4}^{2} ; (t_{1} t_{2} t_{3}^{2}m_{1} − 2 t_{1} t_{2} t_{3} t_{4} + t_{4}^{2}t_{1} + t_{3} t_{4}^{2}) t_{1} .

The following are the components of α^{212121} applied to y_{x2x2xx}(t_{1}, t_{2}, m_{1}, t_{3}, t_{4}).

a = − (m_{1} − t_{4} )^{−1} ,
b = \frac{m_{1} − t_{4}}{t_{1} m_{1} − t_{2} − t_{4} t_{1} − t_{4} t_{3}} ,

c = −  (t_{1} m_{1} − t_{2} − t_{4} t_{1} − t_{4} t_{3} )^{3} ; (m_{1} − t_{4} ) (t_{2}^{3} + 3 t_{2}^{2}t_{3} t_{4} + 3 t_{2} t_{3}^{2}t_{4}^{2} + t_{4}^{2}m_{1} t_{3}^{3}),
d = −\frac{t_{2}^{3} + 3 t_{2}^{2}t_{3} t_{4} + 3 t_{2} t_{3}^{2}t_{4}^{2} + t_{4}^{2}m_{1} t_{3}^{3}}{(t_{1} m_{1} − t_{2} − t_{4} t_{1} − t_{4} t_{3} ) (t_{1} t_{2}^{2} + 2 t_{1} t_{2} t_{3} t_{4} + t_{4} t_{3}^{2}t_{1} m_{1} − t_{2} t_{3}^{2}t_{4} )},

e = − (t_{1} t_{2}^{2} + 2 t_{1} t_{2} t_{3} t_{4} + t_{4} t_{3}^{2}t_{1} m_{1} − t_{2} t_{3}^{2}t_{4} )^{3} ; (t_{2}^{3} + 3 t_{2}^{2}t_{3} t_{4} + 3 t_{2} t_{3}^{2}t_{4}^{2} + t_{4}^{2}m_{1} t_{3}^{3}) t_{2}^{3}t_{3}^{3}t_{4} ,
f = −  t_{2} t_{3}^{2}t_{4} ; (t_{1} t_{2}^{2} + 2 t_{1} t_{2} t_{3} t_{4} + t_{4} t_{3}^{2}t_{1} m_{1} − t_{2} t_{3}^{2}t_{4} ) t_{1} .

We now substitute in values for the t_{i} and the m_{i} in order to obtain points in the D_{σ}^{−}(h) on which α^{212121} is defined. In the first three cases we set t_{1} = ±1, t_{2} = ±2, m_{1} = 3, m_{2} = 5, and in the last 4 cases, we set t_{1} = ±1, t_{2} = ±2, t_{3} = ±3, t_{4} = ±5, and m_{1} = 7. In this way, we obtain the connected component for each D_{σ}^{−}(h). To denote a choice of sign we write a list of 6 symbols, with + or − indicating the sign associated to an element j of I(σ), 0 indicating an element of J(σ), and ∗ indicating elements of K(σ).

1. y_{1x12x2}(t_{1}, m_{1}, t_{2}, m_{2}).
Sign choice Signs of a, b, c, d, e, f Connected Component
0 + ∗0 + ∗  − − − + ++  K ⟷ 11
0 + ∗0 − ∗  − − − + −+  J ⟷ 9
0 − ∗0 + ∗  − − + − ++  I ⟷ 10
0 − ∗0 − ∗  − − + − −+  K ⟷ 11

2. y_{xx1x1x}(t_{1}, t_{2}, t_{3}, m_{1}, t_{4}).
Sign choice Signs of a, b, c, d, e, f Connected Component
+ + 0 + ∗+  + − + + +−  H ⟷ 8
+ + 0 + ∗−  − − + + +−  K ⟷ 11
+ + 0 − ∗+  + − + + ++  I ⟷ 10
+ + 0 − ∗−  − − + + ++  F ⟷ 6
+ − 0 + ∗+  + − − + −+  F ⟷ 6
+ − 0 + ∗−  − − − + −+  J ⟷ 9
+ − 0 − ∗+  + − − + −−  K ⟷ 11
+ − 0 − ∗−  − − − + −−  G ⟷ 7
− + 0 + ∗+  + − + − −−  J ⟷ 9
− + 0 + ∗−  − − + − −−  E ⟷ 5
− + 0 − ∗+  + − + − −+  H ⟷ 8
− + 0 − ∗−  − − + − −+  K ⟷ 11
− − 0 + ∗+  + − − − ++  K ⟷ 11
− − 0 + ∗−  − − − − ++  G ⟷ 7
− − 0 − ∗+  + − − − +−  E ⟷ 5
− − 0 − ∗−  − − − − +−  I ⟷ 10

3. y_{1x1xxx}(t_{1}, m_{1}, t_{2}, t_{3}, t_{4}).
Sign choice Signs of a, b, c, d, e, f Connected Component
0 + ∗ + ++  + − − + −+  F ⟷ 6
0 + ∗ + +−  − − − + −+  J ⟷ 9
0 + ∗ + −+  + − + − −+  H ⟷ 8
0 + ∗ + −−  − − + − −+  K ⟷ 11
0 + ∗ − ++  + − + + −+  K ⟷ 11
0 + ∗ − +−  − − + + −+  G ⟷ 7
0 + ∗ − −+  + − − − −+  J ⟷ 9
0 + ∗ − −−  − − − − −+  E ⟷ 5
0 − ∗ + ++  + − − + ++  H ⟷ 8
0 − ∗ + +−  − − − + ++  K ⟷ 11
0 − ∗ + −+  + − + − ++  E ⟷ 5
0 − ∗ + −−  − − + − ++  I ⟷ 10
0 − ∗ − ++  + − + + ++  I ⟷ 10
0 − ∗ − +−  − − + + ++  F ⟷ 6
0 − ∗ − −+  + − − − ++  K ⟷ 11
0 − ∗ − −−  − − − − ++  G ⟷ 7

4. y_{12x21x}(t_{1}, m_{1}, m_{2}, t_{2}).
Sign choice Signs of a, b, c, d, e, f Connected Component
00 + ∗ ∗ +  + − + + +−  H ⟷ 8
00 + ∗ ∗ −  − − + + +−  K ⟷ 11
00 − ∗ ∗ +  + − − + −−  K ⟷ 11
00 − ∗ ∗ −  − − − + −−  G ⟷ 7

5. y_{xxx2x2}(t_{1}, t_{2}, t_{3}, t_{4}, m_{1}).

Sign choice Signs of a, b, c, d, e, f Connected Component
+ + +0 + ∗  − + + + −+  F ⟷ 6
+ + +0 − ∗  − + + + ++  H ⟷ 8
+ + −0 + ∗  − − + + −−  I ⟷ 10
+ + −0 − ∗  − − + + +−  K ⟷ 11
+ − +0 + ∗  − + − − −+  G ⟷ 7
+ − +0 − ∗  − + − + −−  F ⟷ 6
+ − −0 + ∗  − − − + ++  K ⟷ 11
+ − −0 − ∗  − − − + −+  J ⟷ 9
− + +0 + ∗  − + + − ++  K ⟷ 11
− + +0 − ∗  − + + − −+  J ⟷ 9
− + −0 + ∗  − − + − +−  H ⟷ 8
− + −0 − ∗  − − + − −−  E ⟷ 5
− − +0 + ∗  − + − − −−  I ⟷ 10
− − +0 − ∗  − + − − +−  K ⟷ 11
− − −0 + ∗  − − − − −+  E ⟷ 5
− − −0 − ∗  − − − + −−  G ⟷ 7

6. y_{x2x2xx}(t_{1}, t_{2}, m_{1}, t_{3}, t_{4}).

Sign choice Signs of a, b, c, d, e, f Connected Component
+0 + ∗ + +  − − + + −−  I ⟷ 10
+0 + ∗ + −  − + − + −−  F ⟷ 6
+0 + ∗ − +  − + + + −−  K ⟷ 11
+0 + ∗ − −  − − − + −−  G ⟷ 7
+0 − ∗ + +  − − + + ++  F ⟷ 6
+0 − ∗ + −  − + − + ++  J ⟷ 9
+0 − ∗ − +  − + + + ++  H ⟷ 8
+0 − ∗ − −  − − − + ++  K ⟷ 11
−0 + ∗ + +  − − + − +−  H ⟷ 8
−0 + ∗ + −  − + − − +−  K ⟷ 11
−0 + ∗ − +  − + + − +−  E ⟷ 5
−0 + ∗ − −  − − − − +−  I ⟷ 10
−0 − ∗ + +  − − + − −+  K ⟷ 11
−0 − ∗ + −  − + − − −+  G ⟷ 7
−0 − ∗ − +  − + + − −+  J ⟷ 9
−0 − ∗ − −  − − − − −+  E ⟷ 5

7. y_{x21x12}(t_{1}, t_{2}, m_{1}, m_{2}).

Sign choice Signs of a, b, c, d, e, f Connected Component
+00 + ∗∗  − + + + −+  F ⟷ 6
+00 − ∗∗  − + + + −−  K ⟷ 11
−00 + ∗∗  − − − + ++  K ⟷ 11
−00 − ∗∗  − − − + +−  E ⟷ 5

We now have a decomposition of each connected component of B^{∗} into a disjoint union of subsets of form D_{σ}(h). This allows us to calculate the Euler characteristic (for compactly supported cohomology) of each connected component X as an alternating sum:

χ(X) = ∑_{c=0}^{dim(X)}(−1)^{c}n_{c},

where n_{c} is the number of subsets of form D_{σ}(h) of codimension c contained in X (note that the dimension of X is even). Since the components X are smooth (open in B) this compactly supported Euler characteristic coincides with the usual one by Poincaré duality. We note that each D_{σ}(h) has codimension at most 2.

All the required information is displayed in Figure 6, using a format similar to that used in [6], grouping the subsets of form D_{σ}(h) by connected component. We denote each D_{σ}(h) by the sign choice string listed above (first column); note that the number of 0’s gives the codimension.

Remarks 7.1. Fix σ such that D_{σ} is of codimension 2 (cases 1–3). Then, as |I(σ)| = 2 there are 2^{2} = 4 possible choices of sign, giving rise to 4 subsets D_{σ}(h). Two of these lie in component 11, and the other two lie in either components 5 and 6, components 7 and 8, or components 9 and 10 respectively.

Fix σ such that D_{σ} is of codimension 1 (cases 4–7). Then, as |I(σ)| = 4 there are 2^{4} = 16 possible choices of sign, giving rise to 16 subsets D_{σ}(h). Each connected component 5–10 contains precisely two of these subsets, while connected component 11 contains precisely 4 of these.

Theorem 7.2. The Euler characteristic of each connected component of B^{∗} is given in Table II. We also give, for each connected component, and for codimension 0, 1, and 2, the number of subsets of the form D_{σ}(h) of that codimension contained in that component. We thus conclude that there are 10 connected components of Euler characteristic 1 and one of Euler characteristic 2, confirming a total Euler characteristic of B^{∗} of 12, the sum of these.

Table II. Euler characteristics of the connected components.

Connected component Codim 0 Codim 1 Codim 2 Euler characteristic
1  1  0  0  1
2  1  0  0  1
3  1  0  0  1
4  1  0  0  1
5  8  8  1  1
6  8  8  1  1
7  8  8  1  1
8  8  8  1  1
9  8  8  1  1
10  8  8  1  1
11  12  16  6  2

1  + + + + ++  − − − − −−  2
3  − + − + −+  + − + − +−  4

5  \begin{matrix}+ + − + −+ & − − + + −+ \\ − + − − −+ & − − − − ++ \\ − + − + +− & − − − + −− \\ − + + − −− & + − − − −− \\ − + 0 + ∗− & − − 0 − ∗+ \\ 0 + ∗ − −− & 0 − ∗ + −+ \\ − + −0 − ∗ & − − −0 + ∗ \\ −0 + ∗ − + & −0 − ∗ − − \end{matrix} ; −00 − ∗∗  \begin{matrix}− − + − +− & + + − − +− \\ + − + + +− & + + + + −− \\ + − + − −+ & + + + − ++ \\ + − − + ++ & − + + + ++ \\ + + 0 − ∗− & + − 0 + ∗+ \\ 0 + ∗ + ++ & 0 − ∗ − +− \\ + + +0 + ∗ & + − +0 − ∗ \\ +0 + ∗ + − & +0 − ∗ + + \end{matrix} ; +00 + ∗∗  6

7  \begin{matrix}+ − + − ++ & + − + + −− \\ + − − − +− & + + − − −− \\ − + + − +− & − − + − −− \\ − − − + +− & − − − − −+ \\ + − 0 − ∗− & − − 0 + ∗− \\ 0 + ∗ − +− & 0 − ∗ − −− \\ + − +0 + ∗ & − − −0 − ∗ \\ +0 + ∗ − − & −0 − ∗ + − \end{matrix} ; 00 − ∗ ∗ −  \begin{matrix}− + − + −− & − + − − ++ \\ − + + + −+ & − − + + ++ \\ + − − + −+ & + + − + ++ \\ + + + − −+ & + + + + +− \\ + + 0 + ∗+ & − + 0 − ∗+ \\ 0 + ∗ + −+ & 0 − ∗ + ++ \\ + + +0 − ∗ & − + −0 + ∗ \\ +0 − ∗ − + & −0 + ∗ + + \end{matrix} ; 00 + ∗ ∗ +  8

9  \begin{matrix}− + − + ++ & + − − + +− & − + + + +− & − − − + −+ \\ + − + − −− & − + + − −+ & + − − − −+ & + + + − +− \\ + − 0 + ∗− & − + 0 + ∗+ & 0 + ∗ + +− & 0 + ∗ − −+ \\ + − −0 − ∗ & − + +0 − ∗ & +0 − ∗ + − & −0 − ∗ − + \end{matrix} ; 0 + ∗0 − ∗

10  \begin{matrix}+ + + + −+ & + + − + −− & + + − − ++ & + − + + ++ \\ − − − − +− & − − + − ++ & − − + + −− & − + − − −− \\ + + 0 − ∗+ & − − 0 − ∗− & 0 − ∗ + −− & 0 − ∗ − ++ \\ + + −0 + ∗ & − − +0 + ∗ & +0 + ∗ + + & −0 + ∗ − − \end{matrix} ; 0 − ∗0 + ∗

11  \begin{matrix}− − − + ++ & − + + − ++ & + − − − ++ & + − + + −+ \\ − − + − −+ & + + − − −+ & + + + − −− & + − − + −− \\ − + + + −− & − + − − +− & + + − + +− & − − + + +− \\ + + 0 + ∗− & + − 0 − ∗+ & − + 0 − ∗− & − − 0 + ∗+ \\ 0 + ∗ + −− & 0 + ∗ − ++ & 0 − ∗ + +− & 0 − ∗ − −+ \\ + + −0 − ∗ & + − −0 + ∗ & − + +0 + ∗ & − − +0 − ∗ \\ +0 + ∗ − + & +0 − ∗ − − & −0 + ∗ + − & −0 − ∗ + + \\ +00 − ∗∗ & −00 + ∗∗ & 00 + ∗ ∗ − & 00 − ∗ ∗ + \end{matrix} ; 0 + ∗0 + ∗ 0 − ∗0 − ∗

Figure 6. The connected components of B^{∗} with their cell decompositions.

References

1. A. Berenstein and A. Zelevinsky, Total positivity in Schubert varieties, Comment. Math. Helv. 72 (1997), 128–166.

2. V. Deodhar, On some geometric aspects of Bruhat orderings. I. a finer decomposition of bruhat cells, Invent. Math. (1985), no. 79, 499–511.

3. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. (1979), no. 53, 165–184.

4. G. Lusztig, Appendix: a comparison of two graphs, I. M. R. N. 13 (1997), 639–640, Appendix to [6].

5. K. Rietsch, The intersection of opposed big cells in real flag varieties, Proc. R. Soc. Lond. A 453 (1997), 785–791.

6. , Intersections of Bruhat cells in real flag varieties, I. M. R. N. 13 (1997), 623–640.

7. B. Shapiro, M. Shapiro, and A. Vainshtein, Connected components in the intersection of two open opposite Schubert cells in sl_{n}(r)/B, Internat. Math. Res. Notices 1997, no. 10, 469–493.

8. , Skew-symmetric vanishing lattices and intersections of Schubert cells, Internat. Math. Res. Notices 1998, no. 11, 563–588.

9. T. A. Springer, Linear algebraic groups, second edition, Progress in Mathematics, vol. 9, Birkhäuser, Boston, 1998.

University of Leicester

   E-mail address: R.Marsh@mcs.le.ac.uk

DPMMS, Cambridge University and Newnham College

   E-mail address: rietsch@dpmms.cam.ac.uk