Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 34 (1993), No. 2, 257-276.

Specht Modules for Weyl Groups

Sait Halıcıoğlu and A. O. Morris

Department of Mathematics, Ankara University 06100 Tandoğan Ankara, Turkey Department of Mathematics, The University of Wales Aberystwyth SY23 3BZ, United Kingdom

1. Introduction

Over fields of characteristic zero, there are well known construction of the irreducible representations and of irreducible modules, called Specht modules for the symmetric groups S_{n} which are based on elegant combinatorial concepts connected with Young tableaux etc. (see, e.g. [10]). James [8] extended these ideas to construct irreducible representations and modules over arbitrary field. Al-Aamily, Morris and Peel [1] showed how this construction could be extended to deal with the Weyl groups of type B_{n}. In [11] the second author described a possible extension of James’ work for Weyl groups in general, where Young tableaux are interpreted in terms of root systems. We now modify these results and give an alternative generalisation of James’ work which is an improvement and extension of the original approach suggested by Morris.

2. Some General Results On Weyl Groups

In this section we establish the notation and state some results on Weyl groups which are required later . Standard references for this material are N. Bourbaki [3], R. W. Carter [4] and J. E. Humphreys [7] .

Let Φ be a root system in an l-dimensional real space V and π be a simple system in Φ with corresponding positive system Φ^{+} and negative system Φ^{−} . Let

W = W(Φ) =< τ_{α} | α ∈ Φ >

be the Weyl group of Φ, where τ_{α} is the reflection corresponding to α. 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.

We note the following facts which are required later.

2.1 There are | W | simple systems (positive systems) in Φ given by wπ (wΦ^{+}), w ∈ W. The group W acts transitively on the set of simple systems.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1993 Heldermann Verlag, Berlin

2.2 Let Γ be the Dynkin diagram (or Coxeter graph) of Φ. A Weyl group is irreducible if its Dynkin diagram is connected. Irreducible Weyl 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}. For example W(A_{l}) ≅ S_{l+1}, the symmetric group on the set {1, 2, …, l + 1} . As our aim in this paper is to generalise ideas from the symmetric groups, the root system and Dynkin diagram are given in this case. The Dynkin diagram is

and if {ϵ_{1}, ϵ_{2}, …, ϵ_{l+1}} is the standard basis for R^{l+1} , then

\begin{aligned}π & = \{ α_{1} = ϵ_{1} − ϵ_{2}, α_{2} = ϵ_{2} − ϵ_{3}, …, α_{l} = ϵ_{l} − ϵ_{l+1}\} \\ Φ & = \{ ϵ_{i} − ϵ_{j} | 1 ≤ i, j ≤ l + 1\} \\ Φ^{+} & = \{ ϵ_{i} − ϵ_{j} | 1 ≤ i < j ≤ l + 1\} \end{aligned}

2.3 A subsystem Ψ of Φ is a subset of Φ which is itself a root system in the space which it spans. A subsystem Ψ is said to be additively closed if α, β ∈ Ψ, α + β ∈ Φ, then α + β ∈ Ψ. From now on we assume that Ψ is additively closed subsystem of Φ. A Weyl subgroup W(Ψ) of W corresponding to a subsystem Ψ is the subgroup of W generated by the τ_{α}, α ∈ Ψ.

2.4 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]).

2.5 If w ∈ W and U is the subspace of V composed of all vectors fixed by w, then w is a product of reflections corresponding to roots in the orthogonal complement U^{⊥} of U [4].

2.6 The simple system J of Ψ can always be chosen such that J ⊂ Φ^{+} [12].

2.7 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) and furthermore d_{Ψ} is the element of minimal length in the coset d_{Ψ}W(Ψ) [12].

2.8 We now extend the above to cover some further reflection subgroups of Weyl groups. This work is due to Steinberg and we follow the description given by Carter [5] where the results highlighted below are proved.

Let ρ be a non-trivial symmetry of the Dynkin diagram of Φ. Then there is a unique isometry τ of V such that τ(r) is a positive multiple of ρ(r) = r̄ for all simple roots r ∈ π.

The isometry τ satisfies the conditions :

τ(r) = r̄ if all the roots of Φ have same length
τ(r) = \{ \begin{matrix}\frac{1}{\sqrt{2}} r̄ & if r is short \\ \sqrt{2}r̄ & if r is long \end{matrix} for Φ = B_{2} or F_{4}
τ(r) = \{ \begin{matrix}\frac{1}{\sqrt{3}} r̄ & if r is short \\ \sqrt{3}r̄ & if r is long \end{matrix} forΦ = G_{2}.

Clearly the order of τ as an isometry of V is equal to the order of ρ as a permutation of π. Now let

V^{1} = {v ∈ V | τ(v) = v}.

For each v ∈ V let v^{1} denote the projection of v onto the subspace V^{1} . Then v^{1} is the average of the vectors in the orbit of v under τ.

Since τ(r) is a positive multiple of ρ(r) for all r ∈ π we have

τw_{r}τ^{−1} = w_{ρ(r)}, where r ∈ π.

Let

W^{1} = {w ∈ W | τwτ^{−1} = w},

then W^{1} operates faithfully on V^{1}. Let J be an orbit of π under ρ. Let W(J) be the subgroup of W generated by the elements τ_{r} for r ∈ J. Let w_{0}^{J} be the element of W(J) which transforms every positive root in Φ_{J} into a negative root. Then w_{0}^{J} ∈ W^{1} and W^{1} is generated by the elements w_{0}^{J} for the different ρ-orbits of π. Since w_{0}^{J} coincides with w_{r^{1}} on V^{1} for each root r ∈ J the elements w_{0}^{J} are reflections when restricted to V^{1}. Then the reflections w_{r^{1}} of V^{1}, for all r ∈ π, generate the group W^{1} of isometries of V^{1}. Let Φ^{1} = {r^{1} ∈ V^{1} | r ∈ Φ} and π^{1} = {r^{1} ∈ V^{1} | r ∈ π}. The sets w (Φ_{J}^{+}) form a partition of Φ as w runs through the elements of W^{1} and J runs through the ρ-orbits of π. The roots r and s are in the same set if and only if r^{1} is a positive multiple of s^{1}.

The sets Φ^{1} and π^{1} almost act as a root system and simple system for W^{1} acting on V^{1}, since

(i) Φ^{1} spans V^{1}, (ii) Every element of Φ^{1} is a linear combination of elements of π^{1} with coefficients all

non-negative or all non-positive, (iii) A basis of V^{1} may be obtained by picking one element of π^{1} out of each set of positive

multiples, (iv) If r^{1} ∈ Φ^{1} then there is an element of W^{1} which coincides with w_{r^{1}} on V^{1},

(v) If r^{1}, s^{1} ∈ Φ^{1} then w_{r^{1}}(s^{1}) ∈ Φ^{1}. The groups W^{1} will be referred to as Steinberg subgroups from now on.

3. Specht Modules for Weyl Groups

Let Φ be a root system with simple system π and Dynkin diagram Γ and let Ψ be a

subsystem of Φ with simple system J ⊂ Φ^{+} and Dynkin diagram Δ. If Ψ = ⋃_{i=1}^{r} Ψ_{i}, where Ψ_{i} are the indecomposable components of Ψ , then 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′. If 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^{′⊥}.

Let 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. Let

T_{J,J′} = {wJ̄ | w ∈ W}

Now , we consider under what conditions the elements in the set T_{J,J′} are distinct. We obtain the following lemma.

Lemma 3.1 | T_{J,J′} |=| W | if and only if W(J^{⊥}) ∩ W(J^{′⊥}) =< e >.

Proof. Suppose that w_{1}J̄ = w_{2}J̄, where w_{1}, w_{2} ∈ W. Then w_{1}J̄ = w_{2}J̄, w_{1}J′ = w_{2}J′ and since the elements of each component of J and J′ are also ordered it follows that w_{2}^{−1}w_{1}(α) = α for all α ∈ J and w_{2}^{−1}w_{1}(α) = α for all α ∈ J′, that is, by (2.5) w_{2}^{−1}w_{1} ∈ W(J^{⊥}) ∩ W(J^{′⊥}). Thus we have that the elements of T_{J,J′} are distinct if and only if W(J^{⊥}) ∩ W(J^{′⊥}) =< e >.

Now we can give our principal definition.

Definition 3.2 Let Ψ and Ψ′ be subsystems of Φ with simple systems J and J′ respectively such that Ψ′ ⊆ Φ \ Ψ and J ⊂ Φ^{+}, J′ ⊂ Φ^{+}. The pair {J, J′} is called a useful system in Φ if W(J) ∩ W(J′) =< e > and W(J^{⊥}) ∩ W(J^{′⊥}) =< e >.

Remark 1 If {J, J′} is a useful system in Φ , then {wJ, wJ′} is also a useful system in Φ, for w ∈ W. Thus , from now on T_{J,J′} will be denoted by T_{Δ}.

Remark 2 If {J, J′} is a useful system in Φ then Ψ ∩ Ψ′ = ∅ and Ψ^{⊥} ∩ Ψ^{′⊥} = ∅. However the converse is not true in general.

Definition 3.3 Let {J, J′} be a useful 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 {J, J′} respectively.

We see that this is a natural extension of the concept of a Young tableaux in the representation theory of symmetric groups. In the case A_{l}, the subsystems are given up to conjugacy by root systems of type A_{l_{1}} + ⋯ + A_{l_{s}}, where the l_{i} ≥ 0 and ∑(l_{i} + 1) = l + 1. Thus the corresponding Weyl subgroup is W(A_{l_{1}}) × ⋯ × W(A_{l_{s}}) which is isomorphic to the Young subgroup S_{l_{1}+1} × ⋯ × S_{l_{s}+1} of S_{l+1} and there is a Weyl subgroup corresponding to each partition of l + 1. If Ψ = A_{l_{1}} + ⋯ + A_{l_{s}} is a subsystem of A_{l}, then λ = (l_{1} + 1, …, l_{s} + 1) is a partition of l + 1. If we put k_{i} = l_{1} + ⋯ + l_{i} + i (i = 1, 2, …, s) and k_{0} = 0 then {ϵ_{k_{i−1}+1} − ϵ_{k_{i−1}+2}, …, ϵ_{k_{i}−1} − ϵ_{k_{i}}} is a simple system for A_{l_{i}}. This may be represented by the λ-tableau

1  2  3  . . . . . . k_{1}
k_{1} + 1  k_{1} + 2  k_{1} + 3 . . . . . k_{2}
k_{2} + 1  k_{2} + 2  k_{2} + 3 …. k_{3}
.  .  .  …
k_{s−1} + 1 k_{s−1} + 2 k_{s−1} + 3 . . l + 1

the other n! λ − tableau being obtained by allowing the elements of S_{n} to act on this λ − tableau.

The orthogonal subsystem Ψ^{⊥} is the root system determined by the elements in rows of length one in the λ − tableau. The ordering of the rows is important, for example the 3^{2} − tableaux _{456}^{123} and _{123}^{456} are regarded as distinct tableaux.

Definition 3.4 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 (2.7) we have

τ_{Δ} = {{dJ̄} | d ∈ D_{Ψ}}.

We note that if J̄ = {J; J′} then dJ ⊂ Φ^{+} but dJ′ need not be a subset of Φ^{+}.

We now give an example to illustrate the construction of a Δ-tabloid. In this example and later examples we use the following notation. If π = {α_{1}, α_{2}, …, α_{n}} is a simple

system in Φ and α ∈ Φ, then α = ∑_{i=1}^{n} a_{i}α_{i}, where a_{i} ∈ Z. From now on α is denoted by

a_{1}a_{2} ⋯ a_{n} and τ_{α_{1}}, τ_{α_{2}}, …, τ_{α_{n}} are denoted by τ_{1}, τ_{2}, …, τ_{n} respectively.

Example 3.5 Let Φ = A_{3} with simple system π = {α_{1} = ϵ_{1}−ϵ_{2}, α_{2} = ϵ_{2}−ϵ_{3}, α_{3} = ϵ_{3}−ϵ_{4}}. Let Ψ = 2A_{1} be the subsystem of Φ with simple system J = {100, 001}. Then the Dynkin diagram for Ψ is

Let Ψ′ = 2A_{1} be the subsystem of Φ which is contained in Φ \ Ψ, with simple system J′ = {110, 011}. Since W(J) ∩ W(J′) =< e > and W(J^{⊥}) ∩ W(J^{′⊥}) =< e >, then {J, J′} is a useful system in Φ . Then τ_{Δ} contains the Δ − tabloids (we also give these in the traditional notation as in [9]);

The group W acts on τ_{Δ} according to

σ\{\overline{wJ}\} = \{\overline{σwJ}\} for all σ ∈ W.

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 Δtabloids. 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)].

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 {J, J′} be a useful system in Φ. Let

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

where s is the sign function defined in Section 2 . Then e_{J,J′} is called the generalized Δ-polytabloid associated with J.

If w ∈ W(Φ) , then

wκ_{J′} = ∑_{σ∈W(J′)} s(σ)wσ

= ∑_{σ∈W(J′)} s(σ)(wσw^{−1})w

= { ∑_{σ∈W(wJ′)} s(σ)σ}w.

Hence, for all w ∈ W(Φ), we have

w e_{J,J′} = κ_{wJ′} \{\overline{wJ}\} = e_{wJ,wJ′} .  (3.1)

Let S^{J,J′} be the subspace of M^{Δ} generated by e_{wJ,wJ′} where w ∈ W. Then by (3.1)

S^{J,J′} 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 S^{J,J′} is a cyclic submodule generated by any Δ-polyta-bloid.

The following proposition notes some isomorphisms between Specht modules.

Proposition 3.9 Let {J, J′} be a useful system in Φ. Then we have the following isomorphisms:

(i) If w ∈ W, then S^{J,J′} ≅ S^{wJ,wJ′}. (ii) If w ∈ W(J), then S^{J,J′} ≅ S^{J,wJ′}. (iii) If w ∈ W(J′), then S^{J,J′} ≅ S^{wJ,J′}.

Proof. (i) If w ∈ W, then define T_{w} : S^{J,J′} → S^{wJ,wJ′} by

T_{w}(e_{J,J′}) = w(e_{J,J′}).

Then by (3.1) we have w(e_{J,J′}) = e_{wJ,wJ′} ∈ S^{wJ,wJ′}. Clearly T_{w} is an isomorphism. The statements (ii), (iii) follow easily from (i).

Proposition 3.9 says that a generalized Specht module is dependent only on the Dynkin diagram Δ and Δ′ of J and J′ respectively, thus, from now on it will be denoted by S^{Δ,Δ′}.

A Specht module is spanned by the e_{wJ,wJ′} for all w ∈ W ; the next lemma shows that we need only consider a certain subset of W .

Lemma 3.10 Let {J, J′} be a useful system in Φ . Then S^{Δ,Δ′} is generated by e_{dJ,dJ′} , where d ∈ D_{Ψ′}.

Proof. If w ∈ W, then by (2.7), w = dρ, where d ∈ D_{Ψ′} and ρ ∈ W(J′) and since

ρe_{J,J′} = ∑_{σ∈W(J′)} s(σ)ρσ{J̄} = s(ρ)e_{J,J′} ,

we have

e_{wJ,wJ′} = we_{J,J′} = dρe_{J,J′} = s(ρ)de_{J,J′} = s(ρ)e_{dJ,dJ′} .

Lemma 3.11 Let \{J, J′\} be a useful system in Φ and let d ∈ D_{Ψ}. If \{\overline{dJ}\} appears in e_{J,J′} then it appears only once.

Proof. If σ, σ′ ∈ W(J′) and suppose that σ = dw, σ′ = dw′ where w, w′ ∈ W(J). Then d = σw^{−1} = σ′w^{′−1} and σ^{′−1}σ = w^{′−1}w ∈ W(J) ∩ W(J′) =< e >. Hence we have w = w′ and σ = σ′. Then \{\overline{dJ}\} appears in e_{J,J′} only once.

Corollary 3.14 If {J, J′} be a useful system in Φ, then e_{J,J′} ≠ 0.

Proof. By Lemma 3.11 if \{\overline{σJ}\} appears in e_{J,J′} then all the \{\overline{σJ}\} are different, where σ ∈ W(J′). But \{\{\overline{σJ}\} | σ ∈ W(J′)\} is a linearly independent subset of \{\{\overline{dJ}\} | d ∈ D_{Ψ}\}. If e_{J,J′} = 0 then s(σ) = (−1)^{l(σ)} = 0 for all σ ∈ W(J′). This is a contradiction and so e_{J,J′} ≠ 0.

Remark 3 In [11] the second author has defined a partial dual of a subsystem Ψ simply as a subsystem Ψ′ of Φ which is contained in Φ \ Ψ. However the following lemma shows that the extra condition W(J) ∩ W(J′) =< e > in our definition of a useful system is also necessary. Unfortunately this condition which is a group theoretical one is not easily checked and it would be useful if it could be replaced by a criteria in terms of the root system only.

Lemma 3.13 If there exists w ∈ W(J) ∩ W(J′) such that w has order 2, and s(w) = −1 then e_{J,J′} = 0.

Proof. If w ∈ W(J) ∩ W(J′) and w has order 2, then

(e − w){J̄} = {J̄} − {J̄} = 0

and also {e, w} is a subgroup of W(J′). Thus we can select signed coset representatives σ_{1}, σ_{2}, σ_{3}, …, σ_{s} for {e, w} in W(J′) such that

e_{J,J′} = ∑_{σ∈W(J′)} s(σ)σ{J̄} = (∑_{i=1}^{s} σ_{i})(e − w){J̄} = 0.

Example 3.14 Let Φ = B_{3} and π = {α_{1} = ϵ_{1} − ϵ_{2}, α_{2} = ϵ_{2} − ϵ_{3}, α_{3} = ϵ_{3}}. Let Ψ = 3A_{1} be the subsystem of Φ with simple system J = {α_{1} = ϵ_{1} − ϵ_{2}, α̃ = ϵ_{1} + ϵ_{2}, α_{3} = ϵ_{3}} and let Ψ′ = 3A_{1} be the subsystem of Φ with J′ = {α_{2} = ϵ_{2} − ϵ_{3}, α_{1} + α_{2} + α_{3} = ϵ_{1}, α_{2} + 2 α_{3} = ϵ_{2} + ϵ_{3}}. Then Ψ ∩ Ψ′ = ∅. But

W(J) is given by the elements

{e, τ_{1}, τ_{3}, τ_{1}τ_{3}, τ_{2}τ_{3}τ_{1}τ_{2}τ_{3}τ_{1}τ_{2}, τ_{2}τ_{3}τ_{1}τ_{2}τ_{3}τ_{1}τ_{2}τ_{1}, τ_{3}τ_{2}τ_{3}τ_{1}τ_{2}τ_{3}τ_{1}τ_{2}, τ_{3}τ_{2}τ_{3}τ_{1}τ_{2}τ_{3}τ_{1}τ_{2}τ_{1}},

W(J′) is given by the elements

{e, τ_{2}, τ_{1}τ_{2}τ_{3}τ_{2}τ_{1}, τ_{3}τ_{2}τ_{3}, τ_{3}τ_{2}τ_{3}τ_{1}τ_{2}τ_{3}τ_{1}τ_{2}τ_{1}, τ_{3}τ_{2}τ_{3}τ_{2}, τ_{3}τ_{2}τ_{3}τ_{1}τ_{2}τ_{3}τ_{2}τ_{1}, τ_{1}τ_{2}τ_{3}τ_{1}τ_{2}τ_{1}}.

It follows that w = τ_{3}τ_{2}τ_{3}τ_{1}τ_{2}τ_{3}τ_{1}τ_{2}τ_{1} ∈ W(J) ∩ W(J′) and e_{J,J′} = 0.

Lemma 3.15 Let {J, J1^{′}} and {J, J2^{′}} be useful systems in Φ. If Ψ1^{′} ⊆ Ψ2^{′}, then S^{J,J′_{2}} is a KW-submodule of S^{J,J′_{1}}, where J1^{′} and J2^{′} are simple systems for Ψ1^{′} and Ψ2^{′} respectively.

Proof. Since Ψ1^{′} ⊆ Ψ2^{′}, W(J1^{′}) is a subgroup of W(J2^{′}) and we can select coset repre-

sentatives a_{1}, a_{2}, …, a_{n} for W(J1^{′}) in W(J2^{′}) such that W(J2^{′}) = ⋃_{i=1}^{n} a_{i}W(J1^{′}). Then we

have

e_{J,J′_{2}} = ∑_{σ∈W(J′_{2})} s(σ)σ{J̄}

= ∑_{i=1}^{n} a_{i}s(a_{i}) ∑_{σ∈W(J′_{1})} s(σ)σ{J̄}

=(∑_{i=1}^{n} a_{i}s(a_{i}))e_{J,J′_{1}}.

Now we consider under what conditions S^{Δ,Δ′} is irreducible.

Lemma 3.16 Let {J, J′} be a useful system in Φ and let d ∈ D_{Ψ}. Then the following conditions are equivalent:

(i) \{\overline{dJ}\} appears with non-zero coefficient in e_{J,J′}. (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 = σρ.

Proof. The equivalence of (i) and (ii) follows directly from the formula

e_{J,J′} = ∑_{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}dJ̄ = ρJ̄. Then ρ^{−1}σ^{−1}d ∈ W(J^{⊥}) ∩ W(J^{′⊥}). Since {J, J′} is a useful system in Φ then d = σρ, 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}\}.

Lemma 3.17 Let \{J, J′\} be a useful system in Φ and let d ∈ D_{Ψ}. If \{\overline{dJ}\} appears in e_{J,J′} then dΨ ∩ Ψ′ = ∅.

Proof. Let α ∈ dΨ. If \{\overline{dJ}\} appears in e_{J,J′} then by Lemma 3.16 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′.

Lemma 3.18 Let {J, J′} be a useful system in Φ and let d ∈ D_{Ψ}. Let dΨ ∩ Ψ′ ≠ ∅. Then κ_{J′} \{\overline{dJ}\} = 0.

Proof. If dΨ ∩ Ψ′ ≠ ∅, then let α ∈ dΨ ∩ Ψ′ and so τ_{α} ∈ W(dJ) ∩ W(J′). Thus

(e − τ_{α})\{\overline{dJ}\} = \{\overline{dJ}\} − \{\overline{dJ}\} = 0.

Since {e, τ_{α}} is a subgroup of W(J′) then we can select signed coset representatives σ_{1}, σ_{2}, σ_{3}, …, σ_{s} for {e, τ_{α}} in W(J′) such that

κ_{J′} \{\overline{dJ}\} = (∑_{i=1}^{s} σ_{i})(e − τ_{α})\{\overline{dJ}\} = 0.

The converse of Lemma 3.17 is not true in general, which leads to the following definition.

Definition 3.19 A useful system {J, J′} in Φ is called a good system if dΨ ∩ Ψ′ = ∅ for d ∈ D_{Ψ} then \{\overline{dJ}\} appears with non-zero coefficient in e_{J,J′}.

Lemma 3.20 Let {J, J′} be a good system in Φ and let d ∈ D_{Ψ}.

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

κ_{J′} \{\overline{dJ}\} = s(σ)e_{J,J′} .

Proof. (i) If \{\overline{dJ}\} does not appear in e_{J,J′}, then by definition of a good system we have dΨ ∩ Ψ′ ≠ ∅ and hence by Lemma 3.17 we have κ_{J′} \{\overline{dJ}\} = 0. (ii) Since \{\overline{dJ}\} appears in e_{J,J′} it follows by Lemma 3.16 that there exists σ ∈ W(J′) such that σ\{J̄\} = \{\overline{dJ}\}. Then we have

κ_{J′} \{\overline{dJ}\} = ( ∑_{ρ∈W(J′)} s(ρ)ρ)\{\overline{σJ}\} = s(σ)e_{J,J′} .

Corollary 3.21 Let {J, J′} be a good system in Φ. If m ∈ M^{Δ} then κ_{J′} m is a multiple of e_{J,J′}.

Proof. If m ∈ M^{Δ} then we have

\begin{aligned}m & = ∑_{d∈D_{Ψ}} α_{d}\{\overline{dJ}\}, where α_{d} ∈ K, \\ κ_{J′} m & = ∑_{d∈D_{Ψ}} α_{d}κ_{J′} \{\overline{dJ}\}. \end{aligned}

But by Lemma 3.20 κ_{J′} m = λe_{J,J′}, where λ ∈ K.

We now define a bilinear form < , > on M^{Δ} by setting

< \{J̄_{1}\}, \{J̄_{2}\} >= \{ \begin{matrix}1 & if \{J̄_{1}\} = \{J̄_{2}\} \\ 0 & otherwise. \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 3.22 Let {J, J′} be a good system in Φ. Let U be submodule of M^{Δ}. Then

either S^{Δ,Δ′} ⊆ U or U ⊆ S^{Δ,Δ^{′⊥}}, where S^{Δ,Δ^{′⊥}} is complement of S^{Δ,Δ′} in M^{Δ}.

Proof. If u ∈ U then

< u, e_{J,J′} > =< u, ∑_{σ∈W(J′)} s(σ)σ{J̄} >

= ∑_{σ∈W(J′)} < s(σ)σ^{−1}u, {J̄} >

=< κ_{J′} u, {J̄} > .

But by Corollary 3.21 κ_{J′} u = λe_{J,J′}, for some λ ∈ K. If λ ≠ 0 for some u ∈ U, then

e_{J,J′} ∈ U, that is, S^{Δ,Δ′} ⊆ U. However, if λ = 0 for all u ∈ U, then < u , e_{J,J′} > = 0,

that is, U ⊆ S^{Δ,Δ^{′⊥}}.

We can now prove our principal result.

Theorem 3.23 Let {J, J′} be a good system in Φ . The KW-module

D^{Δ,Δ′} = S^{Δ,Δ′} /S^{Δ,Δ′} ∩ S^{Δ,Δ^{′⊥}}

is zero or irreducible.

Proof. If U is a submodule of S^{Δ,Δ′} then U is a submodule of M^{Δ} and by Theorem 3.22 either S^{Δ,Δ′} ⊆ U in which case U = S^{Δ,Δ′} or U ⊆ S^{Δ,Δ^{′⊥}} and U ⊆ S^{Δ,Δ′} ∩S^{Δ,Δ^{′⊥}}, which completes the proof.

In the case of K = Q or any field of characteristic zero < , > is an inner product and D^{Δ,Δ′} = S^{Δ,Δ′}. Thus if for a subsystem Ψ of Φ a good system {J, J′} can be found, then we have a construction for irreducible KW-modules. Hence it is essential to show for each subsystem that a good system exists which satisfies Definition 3.19.

In the following example, we show how a good system may be constructed in most cases for the Weyl group of type D_{4}. In a future publication, we shall present an algorithm for constructing a good system for certain subsystems; indeed this algorithm will give a good system with additional properties which will lead to the construction of a K-basis for our Specht modules S^{Δ,Δ′}, which correspond to the basis consisting of standard tableaux in the case of symmetric groups.

Example 3.24 Let Φ = D_{4} with simple system

π = {α_{1} = ϵ_{1} − ϵ_{2}, α_{2} = ϵ_{2} − ϵ_{3}, α_{3} = ϵ_{3} − ϵ_{4}, α_{4} = ϵ_{3} + ϵ_{4}}.

Let e, τ_{2}, (τ_{1}τ_{2}), (τ_{1}τ_{3}τ_{2}), (τ_{1}τ_{2}τ_{1}τ_{3}τ_{2}τ_{1}), (τ_{1}τ_{4}τ_{2}), (τ_{1}τ_{2}τ_{1}τ_{4}τ_{2}τ_{1}), (τ_{3}τ_{4}τ_{2}), (τ_{2}τ_{3}τ_{2}τ_{4}τ_{2}τ_{3}), (τ_{1}τ_{3}τ_{4}τ_{2}), (τ_{1}τ_{2}τ_{1}τ_{3}τ_{2}τ_{1}τ_{4}τ_{2}τ_{1}τ_{3}τ_{2}τ_{4}), (τ_{2}τ_{1}τ_{3}τ_{4}τ_{2}), (τ_{2}τ_{1}τ_{4}τ_{2}τ_{1}τ_{3}) be representatives of conjugate classes C_{1}, C_{2}, C_{3}, C_{4}, C_{5}, C_{6}, C_{7}, C_{8}, C_{9}, C_{10}, C_{11}, C_{12}, C_{13} respectively of W(D_{4}). Then we get the character table of W(D_{4}) as shown in Table 1.

Table 1

The non-conjugate subsystems of D_{4} are:

(1)  Ψ_{1} = A_{3} with simple system J_{1} = {1000, 0100, 0010}
(2)  Ψ_{2} = A3^{′} with simple system J_{2} = {1000, 0100, 0001}
(3)  Ψ_{3} = A3^{′′} with simple system J_{3} = {0100, 0010, 0001}
(4)  Ψ_{4} = 4A_{1} with simple system J_{4} = {1000, 1211, 0010, 0001}
(5)  Ψ_{5} = 3A_{1} with simple system J_{5} = {1000, 0010, 0001}
(6)  Ψ_{6} = A_{2} with simple system J_{6} = {1000, 0100}
(7)  Ψ_{7} = 2A_{1} with simple system J_{7} = {1000, 0010}
(8)  Ψ_{8} = 2A1^{′} with simple system J_{8} = {1000, 0001}
(9)  Ψ_{9} = 2A1^{′′} with simple system J_{9} = {0010, 0001}
(10)  Ψ_{10} = A_{1} with simple system J_{10} = {1000}
(11)  Ψ_{11} = D_{4} with simple system J_{11} = {1000, 0100, 0010, 0001}
(12)  Ψ_{12} = ∅ with simple system J_{12} = ∅.

Let Ψ_{1} = A_{3} be the subsystem of D_{4} with J_{1} = {1000, 0100, 0010}. Let Ψ1^{′} = 2A_{1} be the subsystem of Φ which is contained in Φ \ Ψ, with simple system J1^{′} = {1101, 0111}.

Since W(J_{1}) ∩ W(J1^{′}) =< e > and W(J_{1}^{⊥}) ∩ W(J1^{′⊥}) =< e >, then {J_{1}, J1^{′}} is a useful

system in Φ. Then τ_{Δ_{1}} contains the Δ_{1}-tabloids:

\begin{aligned}\{J̄_{1}\} & = \{1000, 0100, 0010; 1101, 0111\} \\ \{τ_{4}J̄_{1}\} & = \{1000, 0101, 0010; 1100, 0110\} \\ \{τ_{2}τ_{4}J̄_{1}\} & = \{1100, 0001, 0110; 1000, 0010\} \\ \{τ_{1}τ_{2}τ_{4}J̄_{1}\} & = \{0100, 0001, 1110; −1000, 0010\} \\ \{τ_{3}τ_{2}τ_{4}J̄_{1}\} & = \{1110, 0001, 0100; 1000, −0010\} \\ \{τ_{1}τ_{3}τ_{2}τ_{4}J̄_{1}\} & = \{0110, 0001, 1100; −1000, −0010\} \\ \{τ_{2}τ_{1}τ_{3}τ_{2}τ_{4}J̄_{1}\} & = \{0010, 0101, 1000; −1100, −0110\} \\ \{τ_{4}τ_{2}τ_{1}τ_{3}τ_{2}τ_{4}J̄_{1}\} & = \{0010, 0100, 1000; −1101, −0111\} \end{aligned}

For d = e, τ_{1}τ_{2}τ_{4}, τ_{3}τ_{2}τ_{4}, τ_{4}τ_{2}τ_{1}τ_{3}τ_{2}τ_{4} we have dΨ_{1} ∩ Ψ1^{′} = ∅. Since

e_{J_{1},J′_{1}} = {J̄} − {τ_{1}τ_{2}τ_{4}J̄} − {τ_{3}τ_{2}τ_{4}J̄} + {τ_{4}τ_{2}τ_{1}τ_{3}τ_{2}τ_{4}J̄}

then {J_{1}, J1^{′}} is a good system in Φ.

Now let K be a field with CharK=0. Let M^{Δ_{1}} be K-space whose basis elements are the Δ_{1}-tabloids. Let S^{Δ_{1},Δ′_{1}} be the corresponding KW-submodule of M^{Δ_{1}}, then by definition of the Specht module we have

S^{Δ_{1},Δ′_{1}} = Sp{e_{J_{1},J′_{1}}, e_{τ_{4}J_{1},τ_{4}J′_{1}}, e_{τ_{2}τ_{4}J_{1},τ_{2}τ_{4}J′_{1}}}

where

\begin{aligned}e_{J_{1},J′_{1}} & = \{J̄_{1}\} − \{τ_{1}τ_{2}τ_{4}J̄_{1}\} − \{τ_{3}τ_{2}τ_{4}J̄_{1}\} + \{τ_{4}τ_{2}τ_{1}τ_{3}τ_{2}τ_{4}J̄_{1}\} \\ e_{τ_{4}J_{1},τ_{4}J′_{1}} & = \{τ_{4}J̄_{1}\} − \{τ_{1}τ_{2}τ_{4}J̄_{1}\} − \{τ_{3}τ_{2}τ_{4}J̄_{1}\} + \{τ_{2}τ_{1}τ_{3}τ_{2}τ_{4}J̄_{1}\} \\ e_{τ_{2}τ_{4}J_{1},τ_{2}τ_{4}J′_{1}} & = \{τ_{2}τ_{4}J̄_{1}\} − \{τ_{1}τ_{2}τ_{4}J̄_{1}\} − \{τ_{3}τ_{2}τ_{4}J̄_{1}\} + \{τ_{1}τ_{3}τ_{2}τ_{4}J̄_{1}\}. \end{aligned}

Let T_{1} be the matrix representation of W afforded by S^{Δ_{1},Δ′_{1}} with character ψ_{1} and let τ_{2} be the representative of the conjugate class C_{2}. Then

\begin{aligned}τ_{2}(e_{J_{1},J′_{1}}) & = \{J̄_{1}\} − \{τ_{1}τ_{2}τ_{4}J̄_{1}\} − \{τ_{3}τ_{2}τ_{4}J̄_{1}\} + \{τ_{4}τ_{2}τ_{1}τ_{3}τ_{2}τ_{4}J̄_{1}\} = e_{J_{1},J′_{1}} \\ τ_{2}(e_{τ_{4}J_{1},τ_{4}J′_{1}}) & = \{τ_{2}τ_{4}J̄_{1}\} − \{τ_{1}τ_{2}τ_{4}J̄_{1}\} − \{τ_{3}τ_{2}τ_{4}J̄_{1}\} + \{τ_{1}τ_{3}τ_{2}τ_{4}J̄_{1}\} = e_{τ_{2}τ_{4}J_{1},τ_{2}τ_{4}J′_{1}} \\ τ_{2}(e_{τ_{2}τ_{4}J_{1},τ_{2}τ_{4}J′_{1}}) & = \{τ_{4}J̄_{1}\} − \{τ_{1}τ_{2}τ_{4}J̄_{1}\} − \{τ_{3}τ_{2}τ_{4}J̄_{1}\} + \{τ_{2}τ_{1}τ_{3}τ_{2}τ_{4}J̄_{1}\} = e_{τ_{4}J_{1},τ_{4}J′_{1}} \end{aligned}

Thus we have

T_{1}(τ_{2}) = ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ) and ψ_{1}(τ_{2}) = 1.

Table 2

By a similar calculation to the above it can be shown that ψ_{1} = χ_{4}. By the same method to the above, we get Table 2. We note that the irreducible modules corresponding to the characters χ_{10}, χ_{11}, χ_{12} have not been obtained. We now show how an additional irreducible character is obtained. Let Ψ2^{′} = A_{1} be the subsystem of Φ with simple system J2^{′} = {1101}. Then {J_{1}, J2^{′}} is a useful system in Φ. Since Ψ2^{′} ⊂ Ψ1^{′}, by Lemma 3.15 S^{Δ_{1},Δ′_{1}} is a KW-submodule of S^{Δ_{1},Δ′_{2}}. By a similar calculation to the above it can be shown that the corresponding character of W afforded by S^{Δ_{1},Δ′_{2}}/S^{Δ_{1},Δ′_{1}} is χ_{11}.

In fact, we have a Specht series corresponding to series

∅ ⊂ {1101} ⊂ {1101, 0111}.

Unfortunately, no further irreducible character of W(D_{4}) are obtained by similar calculations for the other subsystems.

We see that not all the irreducible modules for W(D_{4}) are obtained in this way. That is, there are not a sufficient supply of subsystems or Weyl subgroups to give a complete set of irreducible modules. In the next section, it will be shown that the irreducible character χ_{10}, χ_{12} of degree 4 and 6 respectively can be obtained by considering W(B_{3}) and W(G_{2}) as Steinberg subgroups of W(D_{4}).

4. Additional Specht Modules for Weyl Groups

In this section we consider our groups as reflection groups and thus, include reflection subgroups these groups. We modify the algorithm for determining subsystems of Φ in (2.4) so as to include Steinberg subgroups of the type described in (2.8). Detailed proofs are not always included as they are either modifications of those in Section 3 or in the earlier paper [11]. In this case, it is necessary to express our results in terms of the subgroups rather than the root subsystems as in Section 3.

Algorithm 4.1:

(i) Form the extended Dynkin diagram of Φ. This is obtained by adding one further node to the graph of Φ corresponding to the negative of the highest root,

(ii) Delete one or more nodes in all possible ways from the extended Dynkin diagram, (iii) Let ρ be a symmetry of the remaining Dynkin diagram. Then follow (2.8) to form a

Steinberg subgroup for this remaining Dynkin diagram. (iv) Take also the duals of the diagrams obtained in the same way from the dual system Φ̃

which is obtained from by interchanging long and short roots,

(v) Repeat the process with the Dynkin diagram obtained and continue any number of times.

Definition 4.2 An extended subsystem Ψ is the root system corresponding to the Dynkin diagram obtained by Algorithm 4.1.

Let Ψ be an extended subsystem of Φ with simple system J and Dynkin diagram Δ. By Algorithm 4.1 we have

(i) If the symmetry ρ is the identity, then W(J) is a Weyl subgroup of W(Φ). (ii) If ρ is a non-trivial symmetry, then W(J) is a Steinberg subgroup of W(Φ).

Definition 4.3 If Ψ is an extended subsystem of Φ, then the reflection subgroup W(J) of W which is generated by the τ_{α}, α ∈ Ψ is called a Steinberg-Weyl (SW)-subgroup of W.

We now define equivalence relation on the elements of W.

w′ ∼ w if and only if w′w^{−1} ∈ W(J) for w, w′ ∈ W(Φ).

Definition 4.4 Let w ∈ W . The equivalence class

{wW(J)} = {w′ | w′ ∼ w}

is called a Δ-tabloid.

Let τ_{Δ} be the set of all Δ-tabloids. In this new setting it is clear that the elements of τ_{Δ} are the left coset representatives of W(J) in W and the number of distinct elements in τ_{Δ} is [W : W(J)].

The group W acts on τ_{Δ} according to

w′{wW(J)} = {w′wW(J)} for all w′ ∈ W.

This action is again easily seen to be well defined.

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

Theorem 4.5 The KW-module M^{Δ} is the permutation module on the SW-subgroup W(J) M^{Δ} is a cyclic KW-module generated by any one tabloid and dim_{K}M^{Δ} = [W : W(J)].

Now we can consider the possibility of constructing a KW-module S^{J,J′} which generalises the Specht modules described in Section 3. In this direction we first define a useful dual of a SW-subgroup W(J).

Definition 4.6 A useful dual of W(J) is a SW-subgroup W(J′) of W(Φ) which satisfies W(J) ∩ W(J′) =< e > .

Then we have the following lemma.

Lemma 4.7 If W(J′) is a useful dual of W(J) then W(w J′) is also a useful dual of W(J) for all w ∈ W(J).

Definition 4.8 Let W(J) be a SW-subgroup of W(Φ) and W(J′) a useful dual of W(J). Let

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

where s is the sign function defined in Section 2. Then e_{J,J′} is called a generalized Δpolytabloid.

Let S^{J,J′} be the subspace of M^{Δ} spanned by all the generalized Δ-polytabloids e_{wJ,wJ′},

w ∈ W. By the same method as in Section 3, S^{J,J′} is a KW-submodule of M^{Δ}, which is called a generalized Specht module.

If W(J′) =< e > then S^{J,J′} ≅ M^{Δ}.

Theorem 4.9 S^{J,J′} is a cyclic submodule generated by any Δ-polytabloid.

Lemma 4.10 Let W(J) be a SW-subgroup of W and let W(J′) be a useful dual of W(J). Let w′ be a left coset representative of W(J′) in W. Then S^{J,J′} is spanned by w′e_{J,J′}.

Lemma 4.11 Let W(J) be a SW-subgroup of W. If {w′W(J)} appears in e_{J,J′} then it appears only once.

Proof. See Lemma 3.11.

Corollary 4.12 If W(J) ∩ W(J′) =< e >, then e_{J,J′} ≠ 0.

Lemma 4.13 Let W(J1^{′}) and W(J2^{′}) be useful duals of W(J). If W(J1^{′}) is a SW-subgroup of W(J2^{′}), then S^{J,J′_{2}} is a KW-submodule of S^{J,J′_{1}}.

Proof. This follows from Lemma 3.15

Lemma 4.14 If w ∈ W and W(J′) is a useful dual of W(J), then the following conditions are equivalent:

(i) {wW(J)} appears with non-zero coefficient in e_{J,J′}. (ii) There exists σ ∈ W(J′) such that σ{W(J)} = {wW(J)}. (iii) There exist ρ ∈ W(J), σ ∈ W(J′) such that w = σρ.

Proof. The equivalence of (i) and (ii) follows directly from the formula

e_{J,J′} = ∑_{σ∈W(J′)} s(σ)σ{W(J)}.

(ii) ⇒ (iii): Suppose that there exists σ ∈ W(J′) such that σ{W(J)} = {wW(J)}. Then we have σ^{−1}w{W(J)} = {W(J)}. By the definition of equivalence, σ^{−1}w ∈ W(J) and there exists ρ ∈ W(J) such that σ^{−1}w = ρ. Hence w = σρ, where σ ∈ W(J′) and ρ ∈ W(J). (iii) ⇒ (ii): If w = σρ, then since ρ ∈ W(J), ρ{W(J)} = {W(J)} and {wW(J)} = {σW(J)}.

Definition 4.15 A useful dual W(J′) of W(J) is called a good dual of W(J) if κ_{J′} {wW(J)} ≠ 0, then {wW(J)} appears in e_{J,J′}.

Lemma 4.16 Let W(J′) be a good dual of W(J).

(i) If {wW(J)} does not appear in e_{J,J′} then κ_{J′} {wW(J)} = 0. (ii) If {wW(J)} appears in e_{J,J′}, then there exists σ ∈ W(J′) such that

κ_{J′} {wW(J)} = s(σ)e_{J,J′} .

Proof. See Lemma 3.20.

Corollary 4.17 If m ∈ M^{Δ}, then κ_{J′} m is a multiple of e_{J,J′}.

We now define a bilinear form < , > on M^{Δ} by setting

< \{w_{1}W(J)\}, \{w_{2}W(J)\} >= \{ \begin{matrix}1 & if w_{1} = w_{2} \\ 0 & otherwise. \end{matrix}

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

Now the analogue of James’ submodule theorem can be proved in this more general setting.

Theorem 4.18 Let U be submodule of M^{Δ}. Then either S^{J,J′} ⊆ U or U ⊆ S^{J,J^{′⊥}} where S^{J,J^{′⊥}} is complement of S^{J,J′} in M^{Δ} .

We can now prove our principal result.

Theorem 4.19 The KW-module D^{J} = S^{J,J′} /S^{J,J′} ∩ S^{J,J^{′⊥}} is zero or irreducible.

Proof. See Theorem 3.23.

Now to illustrate the above we show how a complete system of irreducible modules is determined in the case Φ = D_{4}.

Example 4.20 Let Φ = D_{4} with π = {α_{1} = ϵ_{1} − ϵ_{2}, α_{2} = ϵ_{2} − ϵ_{3}, α_{3} = ϵ_{3} − ϵ_{4}, α_{4} = ϵ_{3} + ϵ_{4}} and let Ψ_{1} = A_{3} be the extended subsystem of D_{4} with simple system J_{1} = {1000, 0100, 0010}. Let Ψ1^{′} = 2A_{1} be the extended subsystem of D_{4} with simple system J1^{′} = {1101, 0111}. Since W(J_{1}) ∩ W(J1^{′}) =< e >, then W(J1^{′}) is a useful dual of W(J_{1}). Then τ_{Δ_{1}} contains the Δ_{1}-tabloids {W(J_{1})}, {τ_{4}W(J_{1})}, {τ_{2}τ_{4}W(J_{1})}, {τ_{1}τ_{2}τ_{4}W(J_{1})}, {τ_{3}τ_{2}τ_{4}W(J_{1})}, {τ_{1}τ_{3}τ_{2}τ_{4}W(J_{1})}, {τ_{2}τ_{1}τ_{3}τ_{2}τ_{4}W(J_{1})}, {τ_{4}τ_{2}τ_{1}τ_{3}τ_{2}τ_{4}W(J_{1})}.

For w = e, τ_{1}τ_{2}τ_{4}, τ_{3}τ_{2}τ_{4}, τ_{4}τ_{2}τ_{1}τ_{3}τ_{2}τ_{4}, we have κ_{J′_{1}}{wW(J_{1})} ≠ 0. Since

e_{J_{1},J′_{1}} = {W(J_{1})} − {τ_{1}τ_{2}τ_{4}W(J_{1})} − {τ_{3}τ_{2}τ_{4}W(J_{1})} + {τ_{4}τ_{2}τ_{1}τ_{3}τ_{2}τ_{4}W(J_{1})},

then W(J1^{′}) is a good dual of W(J_{1}).

Now let K be a field with CharK=0. Let M^{Δ_{1}} be K-space whose basis elements are the Δ_{1}-tabloids. Let S^{J_{1},J′_{1}} be the corresponding KW-submodule of M^{Δ_{1}}, then by definition of the Specht module we have

S^{J_{1},J′_{1}} = Sp{e_{J_{1},J′_{1}}, e_{τ_{4}J_{1},τ_{4}J′_{1}}, e_{τ_{2}τ_{4}J_{1},τ_{2}τ_{4}J′_{1}}}

where

\begin{aligned}e_{J_{1},J′_{1}} & = \{W(J_{1})\} − \{τ_{1}τ_{2}τ_{4}W(J_{1})\} − \{τ_{3}τ_{2}τ_{4}W(J_{1})\} + \{τ_{4}τ_{2}τ_{1}τ_{3}τ_{2}τ_{4}W(J_{1})\} \\ e_{τ_{4}J_{1},τ_{4}J′_{1}} & = \{τ_{4}W(J_{1})\} − \{τ_{1}τ_{2}τ_{4}W(J_{1})\} − \{τ_{3}τ_{2}τ_{4}W(J_{1})\} + \{τ_{2}τ_{1}τ_{3}τ_{2}τ_{4}W(J_{1})\} \\ e_{τ_{2}τ_{4}J_{1},τ_{2}τ_{4}J′_{1}} & = \{τ_{2}τ_{4}W(J_{1})\} − \{τ_{1}τ_{2}τ_{4}W(J_{1})\} − \{τ_{3}τ_{2}τ_{4}W(J_{1})\} + \{τ_{1}τ_{3}τ_{2}τ_{4}W(J_{1})\}. \end{aligned}

Let T_{1} be the matrix representation of W afforded by S^{J_{1},J′_{1}} with character ψ_{1} and let τ_{2} be the representative of the conjugate class C_{2}. Then

τ_{2}(e_{J_{1},J′_{1}}) = {W(J_{1})} − {τ_{1}τ_{2}τ_{4}W(J_{1})} − {τ_{3}τ_{2}τ_{4}W(J_{1})} + {τ_{4}τ_{2}τ_{1}τ_{3}τ_{2}τ_{4}W(J_{1})}
= e_{J_{1},J′_{1}}
τ_{2}(e_{τ_{4}J_{1},τ_{4}J′_{1}}) = {τ_{2}τ_{4}W(J_{1})} − {τ_{1}τ_{2}τ_{4}W(J_{1})} − {τ_{3}τ_{2}τ_{4}W(J_{1})} + {τ_{1}τ_{3}τ_{2}τ_{4}W(J_{1})}
= e_{τ_{2}τ_{4}J_{1},τ_{2}τ_{4}J′_{1}}
τ_{2}(e_{τ_{2}τ_{4}J_{1},τ_{2}τ_{4}J′_{1}}) = {τ_{4}W(J_{1})} − {τ_{1}τ_{2}τ_{4}W(J_{1})} − {τ_{3}τ_{2}τ_{4}W(J_{1})} + {τ_{2}τ_{1}τ_{3}τ_{2}τ_{4}W(J_{1})}
= e_{τ_{4}J_{1},τ_{4}J′_{1}}

Thus we have

T_{1}(τ_{2}) = ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ) and ψ_{1}(τ_{2}) = 1.

By a similar calculation to the above it can be shown that ψ_{1} = χ_{4}. By the same method to the above, in addition to the irreducible modules in Example 3.24 we have,

We note that the irreducible modules corresponding to the characters χ_{10}, χ_{11} have not been obtained. We now show how additional irreducible characters are obtained.

Let Ψ2^{′} = A_{1} be the extended subsystem of Φ with simple system J2^{′} = {1101}. Then W(J2^{′}) is a useful dual of W(J_{1}). Since W(J2^{′}) is a SW-subgroup of W(J1^{′}), by Lemma 4.13 S^{J_{1},J′_{1}} is a KW-submodule of S^{J_{1},J′_{2}}. By a similar calculation to the above it can be shown that the corresponding character of W afforded by S^{J_{1},J′_{2}}/S^{J_{1},J′_{1}} is χ_{11}.

Let Ψ_{2} = A_{1} be the extended subsystem of Φ with simple system J_{2} = {1000}. Let Ψ1^{′} = B_{3} be the extended subsystem of Φ with simple system J1^{′} = \{0100, 0001, \frac{1}{2}(1000 + 1211)} and Ψ2^{′} = A_{3} be another extended subsystem of Φ with simple system J2^{′} = {0100, 0001, 0010}. Then W(J2^{′}) is a useful dual of W(J_{2}) and W(J1^{′}) is a good dual of W(J_{1}). Since W(J2^{′}) is a SW-subgroup of W(J1^{′}), by Lemma 4.13 S^{J_{2},J′_{1}} is a KWsubmodule of S^{J_{2},J′_{2}}. By a similar calculation to the above it can be shown that the corresponding character of W afforded by S^{J_{2},J′_{2}}/S^{J_{2},J′_{1}} is χ_{10}. Thus we have obtained a complete set of irreducible modules for D_{4}.

References

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