Hau-wen Huang and Chih-wen Weng,  Combinatorial Representations of Coxeter Graphs over a Field of Two Elements.

Combinatorial representations of Coxeter groups over a field of two elements ^∗

Hau-wen Huang

Chih-wen Weng

April 14, 2008

Abstract

Let W denote a simply-laced Coxeter group with n generators. We construct an n-dimensional representation φ of W over the finite field F₂ of two elements. The action of φ(W ) on F₂ⁿ by left multiplication is corresponding to a combinatorial structure extracted and general- ized from Vogan diagrams. In each case W of types A, D and E, we determine the orbits of F₂ⁿ under the action of φ(W ), and find that the kernel of φ is the center Z(W ) of W.

1 Introduction

A simply-laced Coxeter group is a group WS(m) with a finite set of generators S ⊆ W_S(m) subject only to relations

(ss⁰)^m(s,s0) = 1,

where m(s, s) = 1 and m(s, s⁰) = m(s⁰, s) ∈ {2, 3} for s 6= s⁰ in S. When m is specified, we write W_S for W_S(m), and if both S and m are specified, we write W for W_S(m). A Coxeter graph S represents a simply-laced Coxeter group WS(m), and vice versa. The vertex set of this graph is S, and there is an edge joining two vertices s and s⁰ whenever m(s, s⁰) = 3. There are

∗Research partially supported by the NSC grant 96-2628-M-009-015 of Taiwan R.O.C..

† ‡Department of Applied Mathematics National Chiao Tung University 1001 Ta Hsueh Road Hsinchu, Taiwan 300, R.O.C..

Coxter groups which are not simple-laced. In this article we always assume simple-laced property in a Coxter group to make the corresponding Coxeter graph S a simple graph.

We shall investigate a kind of flipping puzzle, which is also studied in [10, 11], associated with a given Coxeter graph S. The configuration of the flipping puzzle is S, together with an assignment of a unique state, white or black, on each vertex of S. A move in the puzzle is to select a vertex s which has black state, and then flip the state of each neighbor of s. When S is one of the Dynkin diagrams described in Figure 1, the configuration above is essentially a Vogan diagram with identity involution, which was first defined in [9], in a more general way, as a combinatorial object representing the real form of the corresponding complex simple Lie algebra and a system of choices. See also [1, 2, 5].

An(n ≥ 1) c c c q q q c c c

sn sn−1 sn−2 s3 s2 s1

Dn(n ≥ 4)

c c

c c q q q c c c

"""

bb

sn−1

sn

sn−2 sn−3 s3 s2 s1

E6 c c c c c

c

s5 s4 s3 s2 s1

s6

E7 c c c c c

c

c

s6 s5 s4 s3 s2

s7

s1

E8 c c c c c

c

c c

s7 s6 s5 s4 s3

s8

s2 s1

Figure 1: Simply-laced Dynkin diagrams.

We fix a simply-laced Coxeter group W and its Coxeter graph S, where

|S| = n. Let F₂denote the finite field of two elements 0 and 1. In Section 2, we use the column vector set F₂^S = F₂ⁿto describe the set of configurations in the flipping puzzle associated with S by setting that `_s = 1 iff the configuration

` ∈ F<sub>2</sub><sup>n</sup> has black state in the vertex s. For each vertex s ∈ S, we find a way to associate the move with selecting vertex s as an n × n invertible matrix s over F<sub>2</sub>. This s acts on a configuration ` ∈ F₂ⁿ by left multiplication to become a new configuration s` which has the desired property as stated in the definition of the flipping puzzle when `_s = 1. Unlike in the definition, our move s does not select configuration `, but if a configuration has white state in s, it makes no effect; i.e. if `_s = 0 then s` = `.

Let GL_n(F₂) denote the set of n × n invertible matrices over F₂ and let W denote the subgroup of GL_n(F₂) generated by the moves s for s ∈ S. We refer W to a flipping group of S. In Section 3, we find that the canonical map φ : W → GL_n(F₂), lifted from φ(s) = s for s ∈ S, is a homomorphism with φ(W ) = W. Due to its origination, we refer such a map to the Vogan representation of W. Then we find that the flipping group W has trivial center in Section 4. In Sections 5, 6 and 7, we assume W to be A_n, D_n and E_n respectively. By using the finiteness of W , we can determine the size of the corresponding flipping group W. We find that the kernel of the Vogan representation of W is the center Z(W ) of W when W is finite.

In the flipping puzzle on a Coxeter graph S, two configurations are said to be equivalent if one can be obtained from the other by a sequence of moves.

Let P denote the partition of configurations (i.e. F₂ⁿ) according to the above equivalent relation. As a byproduct of our work, we solve the flipping puzzle associated with S when S is each of A_n, D_n and E_n by determining P. Note that when S is a tree, a generalization of Dynkin diagrams, some partial results on P are obtained in [11] and [10].

2 Flipping groups

Throughout this article, W will be a simply-laced Coxeter group with corre- sponding Coxter graph S of n elements and edge set R = {ss⁰ | m(s, s⁰) = 3}.

We shall construct a matrix group associated with the flipping puzzle on the Coxeter graph S. Let Mat_n(F₂) denote the set of n × n matrices over F₂

with rows and columns indexed by S. Let F₂ⁿdenote the set of n-dimensional column vectors over F₂ indexed by S. For s ∈ S, let es denote the character- istic vector of s in F₂ⁿ; that is es = (0, 0, . . . , 0, 1, 0, . . . , 0)^t, where 1 is in the position corresponding to s.

Definition 2.1. For s ∈ S, we associate a matrix s ∈ Mat_n(F₂), denoted by the bold type of s, as

suv =

½ 1, if u = v, or v = s and uv ∈ R;

0, else, where u, v ∈ S.

The following is a reformulating of Definition 2.1.

Lemma 2.2. For s, v ∈ S,

sev =

( ev, if v 6= s;

e v + P

uv∈R

e

u if v = s.

¤ The flipping puzzle associated with S, which is described in the introduc- tion, is now restated as follows. A configuration is simply a vector ` ∈ F<sub>2</sub><sup>n</sup>, where `_s = 1 (resp. `<sub>s</sub> = 0) means that the vertex s ∈ S has black state (resp. white state). In this setting, if `_s = 1 then s` is the new configuration after the move to select the vertex s. Note that if `_s = 0, we have s` = ` from Lemma 2.2, so we can view the action of s on ` as a feigning move on ` which is not originally defined as a move in the flipping puzzle. The following lemma is immediate from this combinatorial realization.

Lemma 2.3. For s ∈ S, s is an involution; that is s² = I, the identity

matrix. ¤

From Lemma 2.3, s is invertible, so we can give the following definition.

Definition 2.4. Let W denote the subgroup of GL_n(F₂) generated by the set {s | s ∈ S}. W is referring to the flipping group of S.

3 Coxeter groups and their combinatorial rep- resentations

Let W denote a simply-laced Coxeter group. Recall that an n-dimensional representation of W over F₂ is a homomorphism of W into GL_n(F₂). It is notorious difficult in the study of groups only defined by generators and rela- tions. Hence the representation theory of Coxeter groups plays an important role in the study. In [7, Section 5.3], Humphreys gives ”geometric represen- tations” of Coxeter groups and use these representations to show that the finite Coxeter groups are essentially those associated with Dynkin diagrams.

In this section we shall show that the flipping groups defined in the last section give ”combinatorial representations” of simply-laced Coxeter groups.

First we need a lemma.

Lemma 3.1. Let W denote a simply-laced Coxeter group with Coxeter graph S. For s ∈ S, set E_s ∈ Mat_n(F₂) by

E_sev =

( 0,P if v 6= s;

uv∈R

e

u, if v = s for v ∈ S. (3.1) Then with referring to the notation in Definition 2.1, the following (i)-(iii) hold.

(i) s = I + E_s for s ∈ S (ii) E_s⁰E_s= 0, if s⁰s /∈ R.

(iii) If s_is_i−1∈ R for i = 1, 2, . . . , t, then E_stE_st−1· · · E_s0 =

½ E_s0, if s_t = s₀; E_stE_s0, if s_ts₀ ∈ R.

Proof. (i) is immediate from Lemma 2.2. Note that E_s⁰E_sev = 0 by (3.1) for any v, s, s⁰ ∈ S with s⁰s 6∈ R, and hence we have (ii). (iii) follows from the same reason as in (ii) by applying the product of matrices in either side of the equation to ev and obtaining the desired equality in each case.

Theorem 3.2. Let W denote a simply-laced Coxeter group with Coxeter graph S. Let W denote the flipping group of S. Then there exists a surjective homomorphism φ : W → W such that φ(s) = s for s ∈ S. In particular, φ is a representation of W over F₂.

Proof. We have seen s² = I for s ∈ S. It remains to show (ss⁰)² = I if s 6= s⁰ and ss⁰ 6∈ R, and to show (ss⁰)³ = I if ss⁰ ∈ R. For s, s⁰ ∈ S,

ss⁰ = (I + E_s)(I + E_s⁰)

= I + E_s+ E_s⁰ + E_sE_s⁰ by Lemma 3.1(i). In the case s 6= s⁰ and ss⁰ 6∈ R,

(ss⁰)² = (I + E_s+ E_s⁰)(I + E_s+ E_s⁰)

= I + 2E_s+ 2E_s⁰

= I

by Lemma 3.1(ii). In the case ss⁰ ∈ R,

(ss⁰)² = (I + E_s+ E_s⁰ + E_sE_s⁰)(I + E_s+ E_s⁰+ E_sE_s⁰)

= I + 3E_s+ 3E_s⁰ + 4E_sE_s⁰ + E_s⁰E_s

= I + E_s+ E_s⁰+ E_s⁰E_s and

(ss⁰)³ = (ss⁰)²(ss⁰)

= (I + Es+ Es⁰ + Es⁰Es)(I + Es+ Es⁰+ EsEs⁰)

= I + 2E_s+ 4E_s⁰ + 2E_sE_s⁰ + 2E_s⁰E_s

= I by Lemma 3.1(iii).

Definition 3.3. The representation φ defined in Theorem 3.2 is called the Vogan representation of W .

Suppose J ⊆ S. Let W_J denote the subgroup of W generated by the set {s | s ∈ J} and W_J denote simply-laced Coxeter group with the set J of generators with the function m ¹ J × J, the restriction of m to J × J. Note that W_J is isomorphic to the subgroup of W generated by the set {s | s ∈ J}

[7, Section 5.5]. Hence we use the same symbol W_J to express these two isomorphic groups. It makes no confused if the place that W_J appears is also considered. For example, the first W_J in (iii) of the following lemma is in the first meaning and the remaining two W_J are in the second meaning.

Note that W_J, which is different to W_J, is the flipping group on J. Let G[J]

denote the submatrix of G ∈ Mat_n(F₂) with rows and columns indexed by J, and W_J[J] := {G[J] | G ∈ W_J}.

Lemma 3.4. Suppose J ⊆ S. The following (i)-(iii) hold.

(i) W_J[J] = W_J.

(ii) The map ψ : W_J → W_J, defined by ψ(G) = G[J] for G ∈ W_J, is a surjective homomorphism.

(iii) Let φ and φ⁰ denote the Vogan representations of W_S and W_J respec- tively. Then φ⁰ = ψ ◦ φ ¹ W_J. In particular, Ker φ ¹ W_J ⊆ Ker φ⁰. Proof. By Definition 2.1, suv = 0 for s, u ∈ J and v ∈ S − J. By this, each matrix G ∈ W_J has the form

G =

µ A 0 B C

¶

if indices in J are placed in the beginning of rows and columns, where A is a

|J| × |J| matrix, B is an (n − |J|) × |J| matrix, C is an (n − |J|) × (n − |J|) matrix, and 0 is a |J| × (n − |J|) zero matrix. Then (i) and (ii) follow from the following matrix product rule in block form:

µ A 0 B C

¶ µ A⁰ 0 B⁰ C⁰

¶

=

µ AA⁰ 0

BA⁰+ CB⁰ CC⁰

¶ .

Since ψ ◦ φ(s) = s[J] = φ⁰(s) by (i) for all s ∈ J, we see φ⁰ = ψ ◦ φ ¹ W_J, and this implies Ker φ ¹ W_J ⊆ Ker φ⁰.

4 The center of a flipping group

As we shall see in Proposition 6.10 that the Coxeter group of type D_n has nontrivial center when n is even. In this section, we show that the center Z(W) of any flipping group W of a Coxeter graph S is trivial. Therefore, the center Z(W ) of any Coxeter group W is contained in the kernel of the Vogan representation of W. Recall that a Coxeter graph S is disconnected if there is a partition of S = S⁰∪ S⁰⁰ with S⁰, S⁰⁰ 6= ∅ and there is no edge uv ∈ R with u ∈ S⁰ and v ∈ S⁰⁰. In this case the Coxeter group W is isomorphic to the direct product W⁰× W⁰⁰ of the Coxeter groups W⁰ = W_S⁰ and W⁰⁰= W_S⁰⁰. S is connected if S is not disconnected.

Proposition 4.1. Let W denote a simple-laced Coxeter group with Coxeter graph S. Then the center Z(W) of the flipping group W of S is trivial.

Proof. It suffices to assume that S is connected with at least two vertices.

Let Z be an element in the center of W and let u, v be two distinct elements in S. We show that Zvu = 0 to conclude Z = I. Suppose Zvu = 1. On the one hand vZeu 6= Zeu since Zeu has 1 in the vth position. On the other hand, vZeu = Zveu = Zeu since veu = eu. Hence we have a contradiction.

From the above Proposition 4.1 we immediately have the following corol- lary.

Corollary 4.2. Let W denote a simply-laced Coxeter group. Then the center Z(W ) is contained in the kernel of the Vogan representation of W. ¤

5 Coxeter groups of type A

Recall that the Vogan representation φ of W is faithful whenever φ is injective.

Also φ is irreducible if there is no subspace V ⊆ F₂ⁿ, V 6= 0, F₂ⁿ, such that φ(W )V ⊆ V. For a ∈ F₂ⁿ, the subset of F₂ⁿconsisting of all elements Ga with G ∈ φ(W ) is called the orbit of F₂ⁿ containing a under the action of φ(W ).

In this section we assume that W is of type A_n with the Coxeter graph S as shown in Fig. 1, and determine the orbits of F₂ⁿ under the action of φ(W ). We also show that the kernel of the Vogan representation φ of W is the center Z(W ) of W and determine the reducibility of φ. The trivial case is given in the following.

Proposition 5.1. Let W be a Coxeter group of type A₁ with the Vogan representation φ. Then the orbits of F₂ are {0}, {1} under the action of φ(W ), Ker φ = {1, s1} = W = Z(W ), and φ is irreducible.

Proof. This follows from that W = {1, s₁} and φ(W ) is a trivial group.

In the remaining of this section, we always assume n ≥ 2. Set

1 = es₁, i + 1 = s_is_i−1· · · s₁1 for 1 ≤ i ≤ n. (5.1) Note that

i = es_i−1+ es_i for 2 ≤ i ≤ n, (5.2) and

n + 1 = es_n= 1 + 2 + · · · + n. (5.3)

Set ∆ = ∆(A_n) := {1, 2, . . . , n}. Note that ∆ is a basis of F₂ⁿ. We refer ∆ to a simple basis of F₂ⁿ. For a ∈ F₂ⁿ, let ∆(a) denote the subset of ∆ consisting of all the elements appeared in the expression of a as a linear combination of elements in ∆. The weight of an element a ∈ F₂ⁿ is wt(a) := |∆(a)|. For example, ∆(n + 1) = ∆ and wt(n + 1) = n.

Lemma 5.2. s_ii = i + 1, s_ii + 1 = i and s_i fixes other vectors in {1, 2, . . . , n + 1} − {i, i + 1} for 1 ≤ i ≤ n.

Proof. This is immediate by applying Lemma 2.2, (5.1) and (5.2).

Let S_n+1denote the group of permutations on {1, 2, . . . , n + 1}. By Lemma 5.2, we can give the following definition.

Definition 5.3. Let α : W → Sn+1 be the homomorphism defined by α(G)j = Gj

for each 1 ≤ j ≤ n + 1 and G ∈ W.

Note that α(si) is the transposition (i, i + 1) in Sn+1 for each 1 ≤ i ≤ n.

Lemma 5.4. α is an isomorphism from W onto Sn+1.

Proof. α is surjective since the transpositions α(s₁), α(s₂),. . ., α(s_n) generate S_n+1. Since ∆ ∪ {n + 1} spans F₂ⁿ, α is injective.

The next proposition determines the orbits of F₂ⁿ under the action of W.

Proposition 5.5. For 0 ≤ i ≤ bⁿ⁺¹₂ c,

O_i = {a ∈ F₂ⁿ | wt(a) = i or n + 1 − i}

is an orbit of F₂ⁿ under the action of W, where btc is the largest integer less than or equal to t.

Proof. Suppose a ∈ F₂ⁿ with wt(a) = i. Observe that from Lemma 5.4 and (5.3),

∆(Ga) =

½ α(G)∆(a), if n + 1 6∈ α(G)∆(a) ;

∆ − α(G)∆(a), if n + 1 ∈ α(G)∆(a)

for G ∈ W. The proposition follows from this observation because the sub- group of α(W) = S_n+1 generated by the transpositions α(s₁), α(s₂),. . ., α(s_n−1) acts transitively on the fixed size subsets of ∆, and s_nn = 1 + 2 +

· · · + n by Lemma 5.2 and (5.3).

In the following propositions, we study the reducibility of φ and Ker φ.

Proposition 5.6. The Vogan representation φ of W is irreducible if and only if n is even.

Proof. Let V denote a nontrivial proper subspace of F₂ⁿsuch that φ(W )V ⊆ V . Referring to Proposition 5.5, note that

V =[

i∈J

Oi (5.4)

for some proper subset J ⊆ {0, 1, . . . , bⁿ⁺¹₂ c} with J 6= {0}. Note that the set in the right side of (5.4) to be closed under addition is when it is the set of even weight vectors, and this occurs if and only if n is odd.

Proposition 5.7. The Vogan representation φ of W is faithful. In particu- lar, Ker φ = Z(W ) is the trivial group.

Proof. The first statement follows from that Proposition 5.4 and W is isomor- phic to S_n+1[7, p41]. The second follows from the first and Corollary 4.2.

6 Coxeter groups of type D

Fix an integer n ≥ 4. Let W denote the Coxeter group of type Dn with the Coxeter graph in Fig. 1. Let φ denote the Vogan representation of W , and W = φ(W ) be the flipping group of S. Set

1 = es1, i + 1 = sisi−1· · · s11 for 1 ≤ i ≤ n − 1, and n + 1 = esn. (6.1) Note that

i = es_i−1+ es_i for 2 ≤ i ≤ n − 2,

n − 1 = es_n−2+ es_n−1+ es_n, (6.2) and

n = es_n−1+ es_n = 1 + 2 + · · · + n − 1. (6.3) Set ∆ = ∆(Dn) := {1, 2, . . . , n − 1, n + 1} to be the simple basis of F₂ⁿ in the case of type D_n. Set ∆(a) and wt(a) as before for a ∈ F₂ⁿ. For example,

∆(n) = ∆ − {n + 1} by (6.3), and wt(n) = n − 1.

Lemma 6.1. The following (i),(ii) hold.

(i) For each 1 ≤ i ≤ n − 1, s_ii = i + 1, s_ii + 1 = i, and

s_ij = j for j ∈ {1, 2, . . . , n + 1} − {i, i + 1}.

(ii) snn − 1 = n, snn = n − 1, snn + 1 = n − 1 + n + n + 1, and snj = j for j ∈ {1, 2, . . . , n − 2}.

In particular, n + 1 ∈ ∆(Gn + 1) and G({1, 2, . . . , n}) ⊆ {1, 2, . . . , n} for all G ∈ W.

Proof. This follows immediately from Lemma 2.2, (6.1) and (6.2).

Let S_ndenote the group of permutations on {1, 2, . . . , n}. By Lemma 6.1, we can give the following definition.

Definition 6.2. Let β : W → S_n denote the homomorphism defined by β(G)(j) = Gj

for 1 ≤ j ≤ n and G ∈ W.

In fact, β is surjective since the n − 1 transpositions β(s₁), β(s₂), . . . , β(sn−1) generate Sn. Let Z denote the additive group of (n − 1)-dimensional subspace of F₂ⁿ spanned by the set {1, 2, . . . , n − 1}. Note that a ∈ Z iff n + 1 6∈ ∆(a) for a ∈ F₂ⁿ. By Lemma 6.1 and (6.3), Z is closed under the left multiplication of matrices in W.

Proposition 6.3. The Vogan representation φ of W is not irreducible. In

particular φ(W )Z ⊆ Z. ¤

Hence Z is a disjoint union of orbits of F₂ⁿ under the action of W. Note that F₂ⁿ− Z is also a disjoint union of orbits of F₂ⁿ under the action of W.

The following proposition determines all the orbits of F₂ⁿunder the action of W.

Proposition 6.4. The following are orbits of F₂ⁿ under the action of W.

O_i = {a ∈ Z | wt(a) = i or n − i},

Ω_o = {a ∈ F₂ⁿ− Z | wt(a) ≡ 1 or n − 1 (mod 2)}, Ωe = {a ∈ F₂ⁿ− Z | wt(a) ≡ 0 or n (mod 2)},

where 0 ≤ i ≤ bⁿ₂c. In particular Ω_o = Ω_e = F₂ⁿ− Z is an orbit when n is odd.

Proof. The proof is similar to the proof of Proposition 5.5. The reason that O_i is an orbit follows from two facts: (i) β(s₁), β(s₂), . . . , β(s_n−2) generate the subgroup Sn−1of Sn consisting of permutations on ∆ − {n + 1} and Sn−1

acts transitively on fixed size subsets of ∆ − {n + 1}, and (ii) s_n−1n − 1 = s_nn − 1 = n = 1 + 2 + · · · + n − 1

by Lemma 6.1(i),(ii) and (6.3). The reason that Ω_o and Ω_e are orbits follows from an additional fact that

wt(s_nn + 1) = wt(n − 1+n+n + 1) = wt(1+2+· · ·+n − 2+n + 1) = n−1.

We study the structure of W.

Definition 6.5. Let γ : W → Aut(Z) denote the homomorphism from W into the group Aut(Z) of automorphisms of Z such that

γ(G)(u) = Gu for G ∈ W and u ∈ Z.

Lemma 6.6. There exists a homomorphism θ : S_n → Aut(Z) such that γ = θ ◦ β.

Proof. Since β is surjective, it suffices to show that the kernel of β is contained in the kernel of γ. Suppose G ∈ Ker β. Then Gi = i for 1 ≤ i ≤ n;

in particular G fixes each element in the basis ∆ − {n + 1} of Z. Thus G ∈ Ker γ.

Let Z o_θS_n denote the group of external semidirect product of Z and S_n with respect to θ[8, p.155]; i.e. Z o_θS_n is the set Z × S_n with the following product rule:

(u, σ)(v, τ ) = (u + θ(σ)(v), στ ),

where u, v ∈ Z and σ, τ ∈ S_n. Note that n + 1 + Gn + 1 ∈ Z for any G ∈ W by Lemma 6.1.

Definition 6.7. Let δ : W → Z o_θS_n denote the map defined by δ(G) = (n + 1 + Gn + 1, β(G))

for any G ∈ W.

Lemma 6.8. δ is an injective homomorphism of W into Z o_θS_n. Proof. For G, H ∈ W,

δ(G)δ(H) = (n + 1 + Gn + 1, β(G))(n + 1 + Hn + 1, β(H))

= (n + 1 + Gn + 1 + θ(β(G))(n + 1 + Hn + 1), β(G)β(H))

= (n + 1 + Gn + 1 + G(n + 1 + Hn + 1), β(G)β(H))

= (n + 1 + GHn + 1, β(GH))

= δ(GH).

This shows that δ is a homomorphism. δ is injective since if n + 1+Gn + 1 = 0 and G ∈ Ker β , then G fixes all vectors in ∆, so G is the identity matrix.

Note that Z = n + 1 + Ω_o if n is odd, and Z = (n + 1 + Ω_o) ∪ (n + 1 + Ω_e) if n is even.

Lemma 6.9. δ(W) = (n + 1 + Ω_o) o_θS_n. In particular δ(W) = Z o_θS_n if n is odd; δ(W) has index 2 in Z o_θS_n if n is even.

Proof. Note that δ(s1), δ(s2), . . . , δ(sn−2), δ(sn−1) generate 0 oθSn. Since Ωo

is an orbit containing n + 1, we have δ(W) = (n + 1 + Ω_o) o_θS_n. The second part follows from Proposition 6.4.

Proposition 6.10. The Vogan representation φ of W is faithful when n is odd; Ker φ has order 2 when n is even. Moreover, Ker φ is the center Z(W ) of W.

Proof. Note that W is isomorphic to the semidirect product Z o S_n of Z and Sn [7, p.42]. By Lemma 6.9, φ is faithful when n is odd, and Ker φ has order 2 when n is even. From Corollary 4.2, Z(W ) ⊆ Ker φ, and from the fact that a normal subgroup of order 2 is contained in the center, we have Ker φ ⊆ Z(W ).

7 Coxeter groups of type E

Fix an integer n ≥ 6. Let W denote the Coxeter group of type E_n with the Coxeter graph S in Fig. 2. In this section we shall determine the orbits of F₂ⁿ under the action of the flipping group W of S. Restricting the attention

to the case n = 6, 7 or 8 in which W is finite, we show that the kernel of the Vogan representation φ of W is the center Z(W ) of W .

En(n ≥ 6) c c c c c

c

q q q c c c

sn−1 sn−2 sn−3 sn−4 sn−5

sn

s3 s2 s1

Figure 2: The Coxeter graph of type E_n.

Set 1 = es₁, i + 1 = s_is_i−1· · · s₁1 for 1 ≤ i ≤ n − 1 and n + 1 = es_n. Note that

i = es_i+ es_i−1 for 2 ≤ i ≤ n − 3,

n − 2 = esn−3+ esn−2+ esn, (7.1) n − 1 = es_n−2+ es_n−1+ es_n,

n = es_n−1+ es_n.

Set ∆ = ∆(En) := {1, 2, . . . , n} to be the simple basis of F₂ⁿ in this case.

Observe that

n + 1 = 1 + 2 + · · · + n. (7.2) Set ∆(a) and wt(a) as before for a ∈ F₂ⁿ. For example, ∆(n + 1) = ∆ and wt(n + 1) = n.

Lemma 7.1. The following (i),(ii) hold.

(i) For each 1 ≤ i ≤ n − 1, s_ii = i + 1, s_ii + 1 = i, and

sij = j for j ∈ {1, 2, . . . , n + 1} − {i, i + 1}.

(ii) s_nn + 1 = n − 2 + n − 1 + n, s_nn = n − 2 + n − 1 + n + 1, s_nn − 1 = n − 2 + n + n + 1, s_nn − 2 = n − 1 + n + n + 1 and

s_nj = j for 1 ≤ j ≤ n − 3.

Proof. This is immediate by applying Lemma 2.2 and (7.1).

Let S_n denote the group of permutations on ∆ = {1, 2, . . . , n}. Set T := {s₁, s₂, . . . , s_n−1}. Recall that W_T is the subgroup of W generated by {s | s ∈ T }. By Lemma 7.1, we find that the set ∆ is closed under the left multiplication of elements in W_T.

Definition 7.2. Let ² : W_T → S_n denote the homomorphism satisfying

²(G)(j) = Gj for 1 ≤ j ≤ n and G ∈ W_T.

In fact, ² is an isomorphism since ∆ is a spanning set and the n − 1 transpositions ²(s₁), ²(s₂), . . . , ²(s_n−1) generate S_n.

Proposition 7.3. The following are orbits of F₂ⁿ under the action of W.

O₀ = {0},

O₁ = {a ∈ F₂ⁿ | a 6= 0, wt(a) ≡ 1 or n − 2 (mod 4)}, (7.3) O₂ = {a ∈ F₂ⁿ | a 6= 0, wt(a) ≡ 2 or n − 3 (mod 4)},

O₃ = {a ∈ F₂ⁿ | a 6= 0, wt(a) ≡ 3 or n (mod 4)}, O4 = {a ∈ F₂ⁿ | a 6= 0, wt(a) ≡ 0 or n − 1 (mod 4)}.

In particular O1 = O3 when n ≡ 1 (mod 4), O1 = O4 and O2 = O3 when n ≡ 2 (mod 4), O₂ = O₄ when n ≡ 3 (mod 4), and O₁ = O₂ and O₃ = O₄ when n ≡ 0 (mod 4).

Proof. It is clear that O₀ is an orbit. There are four cases to put nonzero vectors a, b in an orbit. (a)wt(a) = wt(b) : This is because ²(W_T) = S_n acts transitively on the fixed size subsets of ∆; (b) wt(b) = n + 3 − wt(a), or n − 1 − wt(a) : This is from (a) and the observation that

wt(s_na) =





n + 3 − wt(a), if |∆(a) ∩ {n, n − 1, n − 2}| = 3;

n − 1 − wt(a), if |∆(a) ∩ {n, n − 1, n − 2}| = 1;

w(a), else

(7.4)

by Lemma 7.1(ii) and (7.2); (c) wt(a) = wt(b) − 4 : This is by applying the first case of (7.4) and then applying the second case of (7.4); and (d) wt(a) = wt(b) + 4 : This is by applying the second case of (7.4) and then the first case of (7.4). The proposition follows from the above cases (a)-(d).

Similar to case of A_n, we determine the reducibility of φ from Proposition 7.3 immediately.

Proposition 7.4. The Vogan representation φ is irreducible if and only if n

is even. ¤

Recall that for a ∈ F₂ⁿ, the isotropy group of a in W is {G ∈ W | Ga = a}, and the cardinality of the orbit of a is equal to the index of the isotropy group of a.

Corollary 7.5. For J := {s₂, s₃, . . . , s_n}. the number |W_J||O₁| divides |W|, where

|O₁| =









2ⁿ⁻¹− (−1)ⁿ⁴2ⁿ⁻²² , if n ≡ 0 (mod 4),

2ⁿ⁻¹, if n ≡ 1 (mod 4),

2ⁿ⁻¹+ (−1)ⁿ⁻²⁴ 2ⁿ⁻²² − 1, if n ≡ 2 (mod 4), 2ⁿ⁻²+ (−1)ⁿ⁻³⁴ 2ⁿ⁻³² , if n ≡ 3 (mod 4).

(7.5)

Proof. Since W_J is a subgroup of the isotropy group of 1, the number

|WJ||O1| divides |W|. Note that by (7.3)

|O1| =

























P

k≡1,2( mod 4) 1≤k≤n

¡_n

k

¢, if n ≡ 0 (mod 4), P

k≡1( mod 2) 1≤k≤n

¡_n

k

¢, if n ≡ 1 (mod 4), P

k≡0,1( mod 4) 1≤k≤n

¡_n

k

¢, if n ≡ 2 (mod 4), P

k≡1( mod 4) 1≤k≤n

¡_n

k

¢, if n ≡ 3 (mod 4),

where ¡_n

k

¢ is the binomial coefficient. From this, we routinely prove (7.5) by induction on n.

We need to quote a lemma.

Lemma 7.6. ([4, Lemma 10.2.11]) If W is of type E7 or E8 then Z(W ) = {1, w₀}, where w₀ is the longest element of W . ¤ Recall that T = {s₁, s₂, . . . , s_n−1} and J = {s₂, s₃, . . . , s_n} when we indi- cate that the Coxeter group W is of type E_n.

Proposition 7.7. The Vogan representation φ of W is faithful if W is of type E₆, and |Ker φ| = 2 if W is of type E₇. Moreover, Ker φ = Z(W ) if W is of type E6 or E7.

Proof. Suppose W is of type E₆. With referring to Corollary 7.5, we have

|O1| = 27. By Lemma 3.4(iii) and Proposition 6.10 (the case D5), we know

|W_J| = 2⁴5!, where J is of type D₅. Since |W_J||O₁| divides |W|, we have

|W| ≥ 2⁴5! · 27 = 2⁷3⁴5. Since |W | = 2⁷3⁴5 [7, p.44], W is isomorphic to W and Ker φ is trivial. By this and Corollary 4.2, Z(W ) is trivial.

Suppose W is of type E₇. From Corollary 4.2 and Lemma 7.6, |Ker φ| ≥ 2.

Since |W | = 2¹⁰3⁴5 · 7 [7, p.44], we see that |W| ≤ 2⁹3⁴5 · 7. On the other hand, according to a similar counting argument as above, we have |O1| = 28,

|W_J| = 2⁷3⁴5, where J is of type E₆, and hence |W| ≥ 2⁹3⁴5 · 7. Thus,

|W| = 2⁹3⁴5 · 7 and |Z(W )| = |Ker φ| = 2.

We now go to the last case W of type E₈. Note that J = {s₂, s₃, . . . , s₈} is of type E₇ and T ∩ J = {s₂, s₃, . . . , s₇} is of type A₆. We need more information of the nontrivial element w₀in the center Z(W_J) of W_J. It is quite complicate to describe w₀ directly as a product of elements in J. We borrow two notations to describe w₀. Let φ denote the Vogan representation of W . Note that φ ¹ W_{T ∩J} is an isomorphism of W_{T ∩J} onto W_{T ∩J} by Lemma 3.4(ii) and Proposition 5.7. Also ² ¹ W_{T ∩J} : W_{T ∩J} → S₇ is an isomorphism , where

² is as in Definition 7.2 and S₇ is the group of permutations on {2, 3, . . . , 8}.

The expression of w₀ is as follows.

w₀ = φ⁻¹(²⁻¹((2, 8, 3, 7, 4, 6, 5)))s₈φ⁻¹(²⁻¹((5, 8)(4, 7)(3, 6)))s₈ φ⁻¹(²⁻¹((4, 8)(3, 7)(2, 6)))s₈φ⁻¹(²⁻¹((5, 8)(4, 7)))s₈ φ⁻¹(²⁻¹((3, 7)(2, 6)))s₈.

(7.6)

It is routine to check that the above w₀ maps to −I by the faithful represen- tation defined in [3, Proposition 8] with c = 0 or in [6, p. 291] to conclude w₀ is in the center of W_J and indeed is the longest element of W_J by [3, Proposition 21]. Thus, we have the following lemma.

Lemma 7.8. Let W be of type E₈ with the Vogan representation φ and w₀ ∈ Z(W_J) be not identity. Then φ(w₀) is

²⁻¹((2, 8, 3, 7, 4, 6, 5))s₈²⁻¹((5, 8)(4, 7)(3, 6))s₈²⁻¹((4, 8)(3, 7)(2, 6))s₈

× ²⁻¹((5, 8)(4, 7))s8²⁻¹((3, 7)(2, 6))s8.

¤

Note that W_J is not isomorphic to its flipping group W_J by Proposi- tion 7.7. The following lemma claims that W_J is isomorphic to the subgroup WJ of W.

Lemma 7.9. Let W be of type E8 with the Vogan representation φ. Then the restriction φ ¹ W_J of φ to J is injective.

Proof. Let φ⁰ : W_J → W_J denote the Vogan representation of W_J. From Lemma 3.4(iii) and Proposition 7.7, we see that Ker φ ¹ W_J ⊆ Ker φ⁰ = {1, w₀}, where w₀ is given in (7.6). To prove that Ker φ ¹ W_J is trivial, it suffices to show that φ(w₀) 6= I. This follows from the computation

φ(w₀)8 = 1 + 8

by applying the expression φ(w₀) in Lemma 7.8 to 8 and using Lemma 7.1 and (7.2) for n = 8 to simplify.

There is a similar result about W of type E₈.

Proposition 7.10. If W is of type E8 then Ker φ has order 2. Moreover, Ker φ = Z(W ).

Proof. We have |O₁| = 2³ · 3 · 5 from (7.5), |W_J| = |W_J| = 2¹⁰3⁴5 · 7 from Lemma 7.9 and |W | = 2¹⁴3⁵5²7[7, p.44]. Therefore, as the proof of Proposi- tion 7.7, Ker φ has order 2 and Ker φ = Z(W ).

8 Concluding remarks

We list the main results of this article as follows.

Dynkin diagram reducibility of φ |Ker φ|

An φ is irr. iff n = 1 or n is even.

½ 2, if n = 1, 1, else.

D_n

(n ≥ 4) φ is not irr.

½ 2, if n is even, 1, else.

E₆ φ is irr. 1

E₇ φ is not irr. 2

E₈ φ is irr. 2

Table 1: The reducibility and the kernel of a Vogan representation φ.

Coxeter graph orbits

A_n O_i= {a ∈ F₂ⁿ | wt(a) = i or n + 1 − i}(0 ≤ i ≤ bⁿ⁺¹₂ c).

D_n (n ≥ 4)

O_i= {a ∈ Z | wt(a) = i or n − i} (0 ≤ i ≤ bⁿ₂c), Ω_o= {a ∈ F₂ⁿ− Z | wt(a) ≡ 1 or n − 1 (mod 2)}, Ω_e= {a ∈ F₂ⁿ− Z | wt(a) ≡ 0 or n (mod 2)}, Ω_o= Ω_e = F₂ⁿ− Z when n is odd.

E_n (n ≥ 6)

O₀ = {0},

O₁ = {a ∈ F₂ⁿ | a 6= 0, wt(a) ≡ 1 or n − 2 (mod 4)}, O₂ = {a ∈ F₂ⁿ | a 6= 0, wt(a) ≡ 2 or n − 3 (mod 4)}, O₃ = {a ∈ F₂ⁿ | a 6= 0, wt(a) ≡ 3 or n (mod 4)}, O₄ = {a ∈ F₂ⁿ | a 6= 0, wt(a) ≡ 0 or n − 1 (mod 4)}.

O₁ = O₃ when n ≡ 1 (mod 4),

O₁ = O₄ and O₂= O₃ when n ≡ 2 (mod 4), O₂ = O₄ when n ≡ 3 (mod 4),

O₁ = O₂ and O₃= O₄ when n ≡ 0 (mod 4).

Table 2: The orbits of F₂ⁿ under the action of the flipping group of a Coxeter graph S.

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Hau-wen Huang

Department of Applied Mathematics National Chiao Tung University 1001 Ta Hsueh Road

Hsinchu, Taiwan 30050, R.O.C.

Email: [email protected] Fax: +886-3-5724679