����������� ������ 56 (2004), 105–114

UDK 512.552

���������� ������ ��� research paper

THE REPRESENTATIONS OF FINITE REFLECTION GROUPS

Muhittin Başer and Sait Halıcıoğlu

Abstract. The construction of all irreducible modules of the symmetric groups over an arbitrary field which reduce to Specht modules in the case of fields of characteristic zero is given by G. D. James. Halıcıoğlu and Morris describe a possible extension of James’ work for Weyl groups in general, where Young tableaux are interpreted in terms of root systems. In this paper, we further develop the theory and give a possible extension of this construction for finite reflection groups which cover the Weyl groups.

1. Introduction

The representation theory of symmetric groups over fields of characteristic zero is well developed and documented with a number of books devoted to the subject. The original approach was due to G. Frobenius and I. Schur followed independently by A.Young in a long series of difficult but highly influential papers. Later, in the 1930’s, W. Specht presented an alternative approach which led in an elegant way to a full set of irreducible modules, now called Specht modules. I. G. Macdonald showed [12] how to obtain irreducible modules for a Weyl groups by a construction using subsystems of the root system of the Weyl groups. Macdonald’s method gives many, but in general not all, of the irreducible modules. In 1976, G. D. James in a very important paper [11], gave an easy and ingenious construction of all the irreducible modules of the symmetric groups over an arbitrary field which reduce to Specht modules in the case of fields of characteristic zero. Al-Aamily, Morris and Peel [1] showed how this construction could be extended to deal with the Weyl groups of type B_{n}. In [13], A. O. Morris described a possible extension of James’ work for Weyl groups in general.

Later, the second author and Morris [7] gave an alternative generalisation of James’ work which is an improvement and extension of the original approach suggested by Morris. In [8], L. Hawkins extended Macdonald’s construction to the case where the subsystem of the roots is replaced by a parabolic subset. Although the conjugacy classes and irreducible characters are known for all finite reflection groups individually no unified approach has been obtained. We now give a possible extension of James’ work for finite reflection groups which is a generalization of the original approach suggested by Halıcıoğlu [6].

AMS Subject Classification: 20 C 33, 20 F 55, 22 E 45

Keywords and phrases: Specht module, tableau, tabloid, finite reflection group.

2. Preliminaries

In this section, we establish the notation and state some results on finite reflection groups which are required later. The basic definitions and background material required here may be found in N. Bourbaki [2], R. W. Carter [3], J. E. Humphreys [9] [10], Grove and Benson [5], Halıcıoğlu and Morris [7].

Let V be l-dimensional Euclidean space over the real field equipped with a positive definite inner product ( , ). For α ∈ V, α ≠ 0, let τ_{α} be the reflection in the hyperplane orthogonal to α, that is, τ_{α} is the linear transformation on V defined by

τ_{α}: V ⟶ V, v ↦ τ_{α}(v) = v − 2 \frac{(α, v)}{(α, α)} α

for all v ∈ V. Let Φ be a root system in V and π be a simple system in Φ with corresponding positive system Φ^{+} and negative system Φ^{−}. Then the finite reflection group

W = W(Φ) = ⟨τ_{α} | τ_{α}^{2} = (τ_{α}τ_{β})^{m_{αβ}} = e, α, β ∈ π and α ≠ β⟩

where e is the identity element of W and m_{αβ} is the order of τ_{α}τ_{β}. Let l(w) denote the length of w and the sign of w, s(w), is defined by s(w) = (−1)^{l(w)}, w ∈ W.

2.1 To each root system Φ, there corresponds a graph Γ called the Coxeter graph (or Dynkin diagram) of W, whose nodes are in one-to-one correspondence with the elements of π. A finite reflection group is irreducible if its Coxeter graph is connected. Finite irreducible reflection groups have been classified and correspond to root systems of type A_{l} (l ≥ 1), B_{l} (l ≥ 2), C_{l} (l ≥ 3), D_{l} (l ≥ 4), E_{6}, E_{7}, E_{8}, F_{4}, G_{2}, H_{3}, H_{4}, I_{2}(p) (p = 5 or p ≥ 7). For example W(A_{l}) ≅ S_{l+1} the symmetric group on the set {1, 2, …, l + 1} and W(G_{2}) ≅ D_{6} dihedral group of order 12.

2.2 A subsystem Ψ of Φ is a subset of Φ which is itself a root system in the space which it spans. The finite reflection subgroup W(Ψ) of W corresponding to a subsystem Ψ is the subgroup of W generated by the τ_{α}, α ∈ Ψ. The subsystems Ψ_{1} and Ψ_{2} are conjugate under W if Ψ_{1} = wΨ_{2} for some w ∈ W.

2.3 The graphs which are Dynkin diagrams of subsystems of Φ may be obtained up to conjugacy by a standard algorithm due independently to E. B. Dynkin, A. Borel and J. de Siebenthal (see e.g. [4]).

2.4 The simple system J of Ψ can always be chosen such that J ⊂ Φ^{+} [14].

2.5 The set D_{Ψ} = { w ∈ W | w(j) ∈ Φ^{+} for all j ∈ J } is a distinguished set of coset representatives of W(Ψ) in W, that is, each element w ∈ W has unique expression of the form d_{Ψ}w_{Ψ}, where d_{Ψ} ∈ D_{Ψ} and w_{Ψ} ∈ W(Ψ). Furthermore d_{Ψ} is the element of minimal length in the coset d_{Ψ}W(Ψ) [3].

2.6 If Ψ is a subsystem of Φ with simple system J ⊂ Φ^{+} and Dynkin diagram Δ then let Ψ = ⋃_{i=1}^{r} Ψ_{i}, where Ψ_{i} are the indecomposable components of Ψ. Let J_{i} be a simple system in Ψ_{i} (i = 1, 2, …, r) and J = ⋃_{i=1}^{r} J_{i}. Let Ψ^{⊥} be the largest subsystem in Φ orthogonal to Ψ and let J^{⊥} ⊂ Φ^{+} the simple system of Ψ^{⊥}. Let Ψ′ be a subsystem of Φ which is contained in Φ \ Ψ, with simple system J′ ⊂ Φ^{+} and Dynkin diagram Δ′, Ψ′ = ⋃_{i=1}^{s} Ψi^{′}, where Ψi^{′} are the indecomposable components of Ψ′, then let Ji^{′} be a simple system in Ψi^{′} (i = 1, 2, …, s) and J′ = ⋃_{i=1}^{s} Ji^{′}. Let Ψ^{′⊥} be the largest subsystem in Φ orthogonal to Ψ′ and let J^{′⊥} ⊂ Φ^{+} the simple system of Ψ^{′⊥}. If J̄ stand for the ordered set {J_{1}, J_{2}, …, J_{r}; J1^{′}, J2^{′}, …, Js^{′}}, where in addition the elements in each J_{i} and Ji^{′} are ordered, then let T_{Δ} = {wJ̄ | w ∈ W}. The pair J̄ = {J; J′} is called a useful system in Φ if W(J) ∩ W(J′) = ⟨e⟩ and W(J^{⊥}) ∩ W(J^{′⊥}) = ⟨e⟩. The elements of T_{Δ} are called Δ-tableaux, the J and J′ are called the rows and the columns of {J; J′} respectively. Two Δ-tableaux J̄ and K̄ are row − equivalent, written J̄ ∼ K̄, if there exists w ∈ W(J) such that K̄ = wJ̄. The equivalence class which contains the Δ-tableau J̄ is {J̄} and is called a Δ-tabloid. Let τ_{Δ} be the set of all Δ-tabloids. Then τ_{Δ} = {{ dJ̄ } | d ∈ D_{Ψ}}. The group W acts on τ_{Δ} as σ\{\overline{wJ}\} = \{\overline{σwJ}\} for all σ ∈ W. Let K be arbitrary field and M^{Δ} be the K-space whose basis elements are the Δ-tabloids. Extend the action of W on τ_{Δ} linearly on M^{Δ}, then M^{Δ} becomes a KW-module. Let

κ_{J′} = ∑_{σ∈W(J′)} s(σ)σ and e_{J,J′} = κ_{J′} {J̄}

where s is the sign function. Then e_{J,J′} is called the generalized Δ-polytabloid

associated with J. Let S^{J,J′} be the subspace of M^{Δ} generated by e_{wJ,wJ′} where

w ∈ W. Then S^{J,J′} is called a generalized Specht module. A useful system {J; J′} in Φ is called a good system if d Ψ ∩ Ψ′ = ∅ for d ∈ D_{Ψ} then \{ \overline{dJ} \} appears with non-zero coefficient in e_{J,J′}. If {J; J′} is a good system, then S^{J,J′} is irreducible [7].

3. Specht modules for finite reflection groups

In this section we show how to construct irreducible modules for finite reflection groups.

By [7], if Ψ_{1} and Ψ_{2} are W-conjugate subsystems of Φ, then the corresponding Specht modules S^{Δ_{1}} and S^{Δ_{2}} are isomorphic. Hence, it is important to choose a representative from the set of W-conjugate subsystems. We now give a natural method to choose the representative.

Let Φ be a root system with simple system π = {α_{1}, α_{2}, …, α_{n}} and α, β ∈ Φ such that α = ∑_{i=1}^{n} a_{i}α_{i} and β = ∑_{i=1}^{n} b_{i}α_{i},

“α ≺ β if and only if for some i, a_{j} = b_{j} for j < i and a_{i} < b_{i}”.

Clearly this is a total ordering on Φ. If Ψ_{1} and Ψ_{2} are W-conjugate subsystems of Φ with simple systems J_{1} = {v_{1}, v_{2}, …, v_{l}} and J_{2} = {u_{1}, u_{2}, …, u_{l}}, respectively, then

“Ψ_{1} ◁ Ψ_{2} if and only if for some i, v_{j} = u_{j} for j < i and v_{i} ≺ u_{i}”.

It is also easy to see that this is a total ordering on the set of W-conjugate subsystems. From now on, we consider the maximum subsystem according to the order ◁ as a representative of W-conjugate subsystems. In the case A_{l}, maximum subsystems are A_{l_{1}} + A_{l_{2}} + ⋯ + A_{l_{s}}, where (l_{1} + 1, l_{2} + 1, …, l_{s} + 1) is a partition of l + 1.

Example 3.1. Let Φ = A_{4} with simple system π = {α_{1} = e_{1} − e_{2}, α_{2} = e_{2} − e_{3}, α_{3} = e_{3} − e_{4}, α_{4} = e_{4} − e_{5}}. In this case, we have three W(A_{4})-conjugate subsystems of type 2A_{1}, that is, Ψ_{1} = 2A_{1}; J_{1} = {α_{1}, α_{3}}, Ψ_{2} = 2A1^{′}; J_{2} = {α_{1}, α_{4}} and Ψ_{3} = 2A1^{′′}; J_{3} = {α_{2}, α_{4}}. The Dynkin diagrams for Ψ_{1}, Ψ_{2}, Ψ_{3} and corresponding compositions

Ψ_{1} = 2A_{1}; J_{1} = {α_{1}, α_{3}}  ◯—⨂ ; —◯— ⨂ λ_{1} = (2, 2, 1)
1 2 3 4
Ψ_{2} = 2A′_{1}; J_{2} = {α_{1}, α_{4}}  ◯—⨂—⨂ ; —◯ λ_{1} = (2, 1, 2)
1 2 3 4
Ψ_{3} = 2A′′_{1}; J_{3} = {α_{2}, α_{4}}  ⨂ ; —◯— ⨂ ; —◯ λ_{1} = (1, 2, 2)
1 2 3 4

Since Ψ_{3} ◁ Ψ_{2} ◁ Ψ_{1}, the maximum subsystem is Ψ_{1} and corresponding composition is λ_{1} = (2, 2, 1) which is a partition of 5.

Let Ψ is a subsystem of Φ with simple system J and Ψ is the maximum subsystem of W-conjugate subsystems. Let J̄ be the ordered set

{J_{1}, J_{2}, …, J_{r}; J1^{′}, J2^{′}, …, Js^{′}},

satisfying (2.6), wJ̄ = {wJ_{1}, wJ_{2}, …, wJ_{r} ; wJ1^{′}, wJ2^{′}, …, wJs^{′}} for w ∈ W and

let T_{Δ} = { wJ̄ | w ∈ W }. Now we can give our principal definition.

Definition 3.2. Let Ψ and Ψ′ be subsystems of Φ with simple system J and J′ respectively. J̄ = {J, J′} is called irreducible system in Φ if

(i) W(J) ∩ W(J′) = ⟨e⟩ and W(J^{⊥}) ∩ W(J′^{⊥}) = ⟨e⟩,

(ii) If dΨ ∩ Ψ′ = ∅ then there exist σ ∈ W(J′) and ρ ∈ W(J) such that d = σρ for d ∈ D_{Ψ}.

By [7], if J̄ is an irreducible system in Φ then |T_{Δ}| = |W|.

Remark 3.3. If J̄ is an irreducible system in Φ and d ∈ D_{Ψ} ∩ D_{Ψ′} then \overline{dJ} is irreducible system in Φ.

Definition 3.4. Let J̄ be an irreducible system in Φ. Then the elements of T_{Δ} are called Δ-tableaux, the J and J′ are called the rows and the columns of J̄, respectively. The subgroups W(J) and W(J′) are called the row group and the column group of J̄, respectively.

Definition 3.5. Two Δ-tableaux J̄ and K̄ are row-equivalent, written J̄ ∼ K̄, if there exists w ∈ W(J) such that K̄ = wJ̄. The equivalence class which contains the Δ-tableau J̄ is {J̄} and is called a Δ-tabloid.

Let τ_{Δ} be the set of all Δ-tabloids. It is clear that the number of distinct elements in τ_{Δ} is [W : W(J)] and by (2.5) we have

τ_{Δ} = {{ dJ̄ } | d ∈ D_{Ψ}}.

The group W acts on τ_{Δ} according to

W (Φ) × τ_{Δ} ⟶ τ_{Δ}, (σ , \{\overline{wJ}\}) ↦ σ\{\overline{wJ}\} = \{\overline{σwJ}\}

This action is well defined, for if \{\overline{w_{1}J}\} = \{\overline{w_{2}J}\}, then there exists ρ ∈ W(w_{1}J) such that \overline{ρw_{1}J} = \overline{w_{2}J}. Hence since σρσ^{−1} ∈ W(σw_{1}J) and \overline{σw_{2}J} = \overline{σρw_{1}J} = (σρσ^{−1})(\overline{σw_{1}J}), we have \{\overline{σw_{1}J}\} = \{\overline{σw_{2}J}\}.

Now if K is arbitrary field, let M^{Δ} be the K-space whose basis elements are the Δ-tabloids. Extend the action of W on τ_{Δ} linearly on M^{Δ}, then M^{Δ} becomes a KW-module. Then we have the following lemma.

Lemma 3.6. The KW-module M^{Δ} is a cyclic KW-module generated by any one tabloid and dim_{K}M^{Δ} = [W : W(J)].

Proof. See [7].

Now we proceed to consider the possibility of constructing a KW-module which corresponds to the Specht module in the case of symmetric groups. In order to do this we need to define a Δ-polytabloid.

Definition 3.7. Let J̄ be an irreducible system in Φ. Let

κJ̄ = ∑_{σ∈W(J′)} s(σ)σ and eJ̄ = κJ̄{J̄}

Then eJ̄ is called the generalized Δ-polytabloid.

If w ∈ W(Φ), then

wκJ̄ = ∑_{σ∈W(J′)} s(σ)wσ = ∑_{σ∈W(J′)} s(σ)(wσw^{−1})w = { ∑_{σ∈W(wJ′)} s(σ)σ}

Hence, for all w ∈ W(Φ), we have

weJ̄ = κ\overline{wJ}\{\overline{wJ}\} = e\overline{wJ}.  (3.1)

Let SJ̄ be the subspace of M^{Δ} generated by e\overline{wJ} where w ∈ W. Then by (3.1) SJ̄ is a KW-submodule of M^{Δ}, which is called a generalized Specht module. Then we have the following theorem.

Theorem 3.8. The KW-module SJ̄ is a cyclic submodule generated by any Δ-polytabloid.

Proof. Straightforward.

Lemma 3.9. Let J̄ be an irreducible system in Φ and let d ∈ D_{Ψ}. If \{\overline{dJ}\} appears in eJ̄ then it appears only once.

Proof. As in Lemma 3.10 [7].

Corollary 3.10. If J̄ be an irreducible system in Φ, then eJ̄ ≠ 0.

Lemma 3.11. Let J̄ be an irreducible system in Φ and let d ∈ D_{Ψ}. Then the following conditions are equivalent:

(i) \{\overline{dJ}\} appears with non-zero coefficient in eJ̄, (ii) there exists σ ∈ W(J′) such that σ\{J̄\} = \{\overline{dJ}\}, (iii) there exists ρ ∈ W(J) and σ ∈ W(J′) such that d = σ ρ, (iv) dΨ ∩ Ψ′ = ∅. Proof. The equivalence of (i) and (ii) follows directly from the formula

eJ̄ = ∑_{σ∈W(J′)} s(σ)σ{J̄}

(ii) ⇒ (iii). Suppose that there exists σ ∈ W(J′) such that σ\{ J̄ \} = \{ \overline{dJ} \}. Then we have σ^{−1} d{ J̄ } = {J̄ }. By the definition of equivalence there exists ρ ∈ W(J) such that σ^{−1} d J̄ = ρ J̄ . Then ρ^{−1}σ^{−1}d ∈ W(J^{⊥}) ∩ W(J^{′⊥}). Since {J, J′} is an irreducible system in Φ then d = σ ρ , where σ ∈ W(J′) and ρ ∈ W(J).

(iii) ⇒ (ii). Let d = σρ, where σ ∈ W(J′) and ρ ∈ W(J). Since ρ ∈ W(J), ρJ̄ = J̄ then \{\overline{dJ}\} = \{\overline{σρJ}\} = \{\overline{σJ}\}.

(i) ⇒ (iv). Let α ∈ dΨ. If \{\overline{dJ}\} appears in eJ̄ then by (i) ⇒ (iii) d = σρ, where σ ∈ W(J′) and ρ ∈ W(J). Then α ∈ σρΨ. Since ρ ∈ W(J), then α ∈ σΨ and σ^{−1}α ∈ Ψ. But Ψ ∩ Ψ′ = ∅, then σ^{−1}α ∉ Ψ′. Since σ ∈ W(J′), σΨ′ = Ψ′ then α ∉ Ψ′.

(iv) ⇒ (i). By definition of irreducible system.

Lemma 3.12. Let J̄ be an irreducible system in Φ and let d ∈ D_{Ψ}. If dΨ∩Ψ′ ≠ ∅, then κJ̄\{\overline{dJ}\} = 0.

Proof. As in Lemma 3.18 [7].

Lemma 3.13. Let J̄ be an irreducible system in Φ and let d ∈ D_{Ψ}.

(i) If \{\overline{dJ}\} does not appear in eJ̄ then κJ̄\{\overline{dJ}\} = 0. (ii) If \{\overline{dJ}\} appears in eJ̄ then there exists σ ∈ W(J′) such that κJ̄\{\overline{dJ}\} =

s(σ)eJ̄.

Proof. See Lemma 3.20 [7].

Corollary 3.14. Let J̄ be an irreducible system in Φ. If m ∈ M^{Δ} then κJ̄m is a multiple of eJ̄.

We now define a bilinear form ⟨ , ⟩ on M^{Δ} by setting

⟨\{J̄\}, \{K̄\}⟩ := \{ \begin{matrix}1, & \{J̄\} = \{K̄\} \\ 0, & \{J̄\} ≠ \{K̄\}. \end{matrix}

This is a symmetric, non-singular, W-invariant bilinear form on M^{Δ}.

Now we can prove James’ submodule theorem in this general setting.

Theorem 3.15. Let J̄ be an irreducible system in Φ. Let U be submodule of M^{Δ}. Then either SJ̄ ⊆ U or U ⊆ SJ̄^{⊥}.

Proof. If u ∈ U then

⟨u, eJ̄⟩ = ⟨u, ∑_{σ∈W(J′)} s(σ)σ{J̄}⟩ = ∑_{σ∈W(J′)}⟨s(σ)σ^{−1}u, {J̄}⟩ = ⟨κJ̄u, {J̄}⟩.

But by Corollary 3.14 κJ̄u = λeJ̄, for some λ ∈ K. If λ ≠ 0 for some u ∈ U, then eJ̄ ∈ U, that is, SJ̄ ⊆ U. However, if λ = 0 for all u ∈ U, then ⟨u, eJ̄⟩ = 0, that is, U ⊆ SJ̄^{⊥}.

We can now prove our principal result.

Theorem 3.16. Let J̄ be an irreducible system in Φ. The KW-module

SJ̄/SJ̄ ∩ SJ̄^{⊥}

is zero or irreducible.

Proof. Let SJ̄/SJ̄ ∩ SJ̄^{⊥} ≠ {0}. We need to show that the KW-module SJ̄/SJ̄ ∩ SJ̄^{⊥} is irreducible. Let U/SJ̄ ∩ SJ̄^{⊥} be a submodule of SJ̄/SJ̄ ∩ SJ̄^{⊥}. Then SJ̄ ∩ SJ̄^{⊥} ⊆ U ⊆ SJ̄ and U = SJ̄ or U = SJ̄ ∩ SJ̄^{⊥} by Theorem 3.15. If U = SJ̄ then U/SJ̄∩SJ̄^{⊥} = SJ̄/SJ̄∩SJ̄^{⊥}. If U = SJ̄∩SJ̄^{⊥} then U/SJ̄∩SJ̄^{⊥} = {0}. Thus SJ̄/SJ̄ ∩ SJ̄^{⊥} is irreducible.

In the case of K = Q or any field of characteristic zero ⟨ , ⟩ is an inner product and SJ̄/SJ̄ ∩ SJ̄^{⊥} ≅ SJ̄. Thus if for a subsystem Ψ of Φ an irreducible system J̄ can be found, then we have a construction for irreducible KW-modules. Hence it is essential to show for each subsystem that an irreducible system exists which satisfies Definition 3.2.

Remark 3.17. For any root system Φ with simple system π, there are two trivial irreducible system in Φ. We can describe immediately the representations arising in these cases.

If J̄_{1} = {π; ∅}, then W(J_{1}) = W(Φ) and W(J_{1}′) = ⟨e⟩. So

eJ̄_{1} = ∑_{σ∈W (J_{1}′)} s(σ)σ{J̄_{1}} = s(e)e{J̄_{1}} = {J̄_{1}}.

We clearly have SJ̄_{1} = Sp{eJ̄_{1}} = Sp{{J̄_{1}}} and weJ̄_{1} = w{J̄_{1}} = {J̄_{1}} = eJ̄_{1} for all w ∈ W and the identity representation of W is given.

If J̄_{2} = {∅; π}, then W(J_{2}) = ⟨e⟩ and D_{Ψ_{2}} = W(Φ) = W(J_{2}′). So

eJ̄_{2} = ∑_{σ∈W (J_{2}′)} s(σ)σ{J̄_{2}} = ∑_{σ∈W(Φ)} s(σ)σ{J̄_{2}}.

Hence SJ̄_{2} = Sp{eJ̄_{2}} and weJ̄_{2} = s(w)eJ̄_{2} for all w ∈ W. Thus w ↦ sgn(w) for all w ∈ W, and the corresponding representation leads to the sign character of W.

In the following example, we show that an irreducible system may be constructed in the case of the finite reflection group of type I_{2}(8).

Example 3.18. Let Φ = I_{2}(8) with simple system π = {α_{1} = e_{1}, α_{2} = cos \frac{7π}{8} e_{1} + sin \frac{7π}{8} e_{2}\} and Φ^{+} = \{α_{1} = e_{1}, α_{2} = cos \frac{7π}{8} e_{1} + sin \frac{7π}{8} e_{2}, α_{3} = cos \frac{π}{8} e_{1} + sin \frac{π}{8} e_{2}, α_{4} = cos \frac{2π}{8} e_{1} + sin \frac{2π}{8} e_{2}, α_{5} = cos \frac{3π}{8} e_{1} + sin \frac{3π}{8} e_{2}, α_{6} = cos \frac{4π}{8} e_{1} + sin \frac{4π}{8} e_{2}, α_{7} = cos \frac{5π}{8} e_{1} + sin \frac{5π}{8} e_{2}, α_{8} = cos \frac{6π}{8} e_{1} + sin \frac{6π}{8} e_{2}\}. Let e, (τ_{1}τ_{2})^{4}, τ_{1}τ_{2}, (τ_{1}τ_{2})^{2}, (τ_{1}τ_{2})^{3}, τ_{1}, τ_{2} be representatives of conjugate classes C_{1}, C_{2}, C_{3}, C_{4}, C_{5}, C_{6}, C_{7} respectively of W(I_{2}(8)). The character table of W(I_{2}(8)) is given by

C_{1}  C_{2}  C_{3}  C_{4}  C_{5}  C_{6}  C_{7}
χ_{1}  1  1  1  1  1  1  1
χ_{2}  1  1  1  1  1  −1  −1
χ_{3}  1  1  −1  1  −1  1  −1
χ_{4}  1  1  −1  1  −1  −1  1
χ_{5}  2  −2  \sqrt{2}  0  −\sqrt{2}  0  0
χ_{6}  2  2  0  −2  0  0  0
χ_{7}  2  −2  −\sqrt{2}  0  \sqrt{2}  0  0

The non-conjugate subsystem of I_{2}(8) are:

1. Ψ_{1} = I_{2}(8); J_{1} = {α_{1}, α_{2}}

2. Ψ_{2} = 2A_{1}; J_{2} = {α_{2}, −α̃}

3. Ψ_{3} = A_{1}; J_{3} = {α_{1}}

4. Ψ_{4} = A1^{′}; J_{4} = {α_{2}}

5. Ψ_{5} = ∅; J_{5} = ∅ where α̃ = cos \frac{3π}{8} e_{1} + sin \frac{3π}{8} e_{2} is the longest root in I_{2}(8).

(1) Let Ψ_{1} = I_{2}(8) be the subsystem of Φ with simple system J_{1} = {α_{1} , α_{2}}. Then Coxeter graph for Ψ_{1} is

⨂—^{8}◯—^{8}◯
− α̃ α_{1} α_{2}

If Ψ1^{′} = ∅ with simple system J1^{′} = ∅, then {α_{1}, α_{2}; ∅} is an irreducible system in Φ by Remark 3.17. Thus we have

C_{1}  C_{2}  C_{3}  C_{4}  C_{5}  C_{6}  C_{7}
ψ_{1}  1  1  1  1  1  1  1

that is the character χ_{1}.

(2) Let Ψ_{2} = 2A_{1} be the subsystem of Φ with simple system J_{2} = {α_{2} , −α̃}. Then Coxeter graph for Ψ_{2} is

◯—^{8}⨂ ; —^{8}◯
− α̃ α_{1} α_{2}

In this case, we have not found an irreducible system in Φ.

(3) Let Ψ_{3} = A_{1} be the subsystem of Φ with simple system J_{3} = {α_{1}}. Then Coxeter graph for Ψ_{3} is

⨂—^{8}◯—^{8}⨂
− α̃ α_{1} α_{2}

If Ψ3^{′} = 2A_{1} with simple system J3^{′} = {α_{4}, α_{8}}, then {α_{1}; α_{4}, α_{8}} is an irreducible system in Φ. Thus we have

C_{1}  C_{2}  C_{3}  C_{4}  C_{5}  C_{6}  C_{7}
ψ_{2}  2  2  0  −2  0  0  0

that is the character χ_{6}.

(4) Let Ψ_{4} = A1^{′} be the subsystem of Φ with simple system J_{4} = {α_{2}}. Then Coxeter graph for Ψ_{4} is

⨂—^{8}⨂ ; —^{8}◯
− α̃ α_{1} α_{2}

If Ψ4^{′} = 2A_{1} with simple system J4^{′} = {α_{4}, α_{8}} then {α_{2}; α_{4}, α_{8}} is an irreducible system in Φ. Thus we have

C_{1}  C_{2}  C_{3}  C_{4}  C_{5}  C_{6}  C_{7}
ψ_{3}  2  2  0  −2  0  0  0

that is the character χ_{6}.

(5) Let Ψ_{5} = ∅ be the subsystem of Φ with simple system J_{5} = ∅. Then Coxeter graph for Ψ_{5} is

⨂—^{8}⨂ ; —^{8}⨂
− α̃ α_{1} α_{2}

If Ψ5^{′} = I_{2}(8) with simple system J5^{′} = {α_{1}, α_{2}} then {∅; α_{4}, α_{8}} is an irreducible system in Φ by Remark 3.17. Thus we have

C_{1}  C_{2}  C_{3}  C_{4}  C_{5}  C_{6}  C_{7}
ψ_{4}  1  1  1  1  1  −1  −1

that is the character χ_{2}.

REFERENCES

[1] E. Al-Aamily, A. O. Morris and M. H. Peel, The representations of the Weyl groups of type B_{n}, J. Algebra, 68 (1981), 298–305.

[2] N. Bourbaki, Groupes et algèbres de Lie, Actualites Sci. Induct 1337, Hermann, Paris, 1968.

[3] R. W. Carter, Simple Groups of Lie Type, Wiley, London, New York, Sydney, Toronto, 1989.

[4] R. W. Carter, Conjugacy classes in the Weyl group, Comp. Math., 25 (1972), 1–59.

[5] L. C. Grove and C. T. Benson, Finite Reflection Groups, Springer-Verlag, 1985.

[6] S. Halıcıoğlu, Specht modules for finite reflection groups, Glasgow Math. J. 37 (1995), 279–287.

[7] S. Halıcıoğlu and A. O. Morris, Specht modules for Weyl groups, Contrib. Alg. Geom., 34 (1993), 257–276.

[8] L. Hawkins, Constructing irreducible representations of Weyl groups, Sem. Lother. Combin., 34 (1995).

[9] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, 1972.

[10] J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, 1990.

[11] G. D. James, The irreducible representations of the symmetric group, Bull. Lond. Math. Soc., 8 (1976), 229–232.

[12] I. G. Macdonald, Some irreducible representation of Weyl groups, Bull. Lond. Math. Soc., 4 (1972), 148–150.

[13] A. O. Morris, Representations of Weyl groups over an arbitrary field, Astèrisque, 87–88 (1981), 267–287.

[14] A. O. Morris and A. J. Idowu, Some combinatorial results for Weyl groups, Proc. Camb. Phil. Soc., 101 (1987), 405–420.

(received 16.03.2004)

M. Başer, Department of Mathematics, Faculty of Science and Arts, Kocatepe University, Afyon - Turkey

E-mail: mbaser@aku.edu.tr

Sait Halıcıoğlu, Department of Mathematics, Faculty of Science, Ankara University, 06100 Ankara - Turkey