James' Submodule Theorem and the Steinberg Module
Meinolf Geck











James' Submodule Theorem and the Steinberg Module
James' Submodule Theorem
 and the Steinberg Module[This paper is a contribution to the Special Issue on the Representation Theory of the Symmetric Groups and Related Topics. The full collection is available at https://www.emis.de/journals/SIGMA/symmetric-groups-2018.htmlhttps://www.emis.de/journals/SIGMA/symmetric-groups-2018.html]
Meinolf GECK
M. Geck
IAZ - Lehrstuhl für Algebra, Universität Stuttgart,

Pfaffenwaldring 57, D–70569 Stuttgart, Germany
mailto:meinolf.geck@mathematik.uni-stuttgart.demeinolf.geck@mathematik.uni-stuttgart.de
<http://www.mathematik.uni-stuttgart.de/ geckmf/>

Received August 29, 2017, in final form November 28, 2017; Published online December 05, 2017

James' submodule theorem is a fundamental result in the representation theory of the symmetric groups and the finite general linear groups. In this note we consider a version of that theorem for a general finite group with a split BN-pair. This gives rise to a distinguished composition factor of the Steinberg module, first described by Hiss via a somewhat different method. It is a major open problem to determine the dimension of this composition factor.

groups with a BN-pair; Steinberg representation; modular representations

20C33; 20C20






§ INTRODUCTION
 
Let G be a finite group of Lie type and _k be the Steinberg
representation of G, defined over an arbitrary field k; see <cit.>.
We shall be concerned here with the case where _k is reducible. There
is only very little general knowledge about the structure of _k in this
case; see, e.g., <cit.> and the references there. Using
the theory of Gelfand–Graev representations of G, Hiss <cit.>
showed that _k always has a certain distinguished composition factor
with multiplicity 1. It appears to be extremely difficult to determine
further properties of this composition factor, e.g., its dimension. The
purpose of this note is to show that this composition factor can be
defined in a somewhat more intrinsic way through a version of James'
submodule theorem <cit.>; see Remark <ref>.



§ GROUPS WITH A SPLIT BN-PAIR
 

Let G be a finite group and B,N⊆ G be subgroups which satisfy
the axioms for an “algebraic group with a split BN-pair” in
<cit.>. We just recall explicitly those properties of G which
will be important for us in the sequel. Firstly, there is a prime number
p such that we have a semidirect product decomposition B=U⋊ H
where H=B∩ N is an abelian group of order prime to p and U is a
normal p-subgroup of B. The group H is normal in N and W=N/H is
a finite Coxeter group with a canonically defined generating set S; let
l W→_0 be the corresponding length function. For each
w∈ W, let n_w∈ N be such that Hn_w=w. Let w_0∈ W be the
unique element of maximal length; we have w_0^2=1. Let n_0∈ N be
a representative of w_0 and V:=n_0^-1Un_0; then U∩ V=H. For
w∈ W, let U_w:=U∩ n_w^-1Vn_w. (Note that V, U_w do not
depend on the choices of n_0, n_w.) Then we have the following sharp
form of the Bruhat decomposition:

    G=∐_w∈ W Bn_wU_w,    ,

that is, every g∈ Bn_wB can be uniquely written as g=bn_wu where b∈ B and u∈ U_w. Note that, since w_0^2=1, we have U_w_0=U
and Bn_0B=Bn_0U=Un_0B, in both cases with uniqueness of expressions.

Let k be any field and kG be the group algebra of G. All our
kG-modules will be finite-dimensional and left modules.

  Let :=∑_b∈ B b∈ kG. Then kG
is a left kG-module which is canonically isomorphic to the induced
module _B^G(k_B); here, k_B denotes the trivial kB-module. Now
kG carries a natural symmetric bilinear form ⟨ , ⟩ kG× kG→ k such that, for any g,g'∈ G, we
have

    ⟨ g,g'⟩=  1,    ,
     0,    .

This form is non-singular and G-invariant. For any subset X⊆
kG, we denote

    X^⊥:={a∈ kG | ⟨ a,x⟩=0 }.

If X is a kG-submodule of kG, then so is X^⊥.


  Let σ U→ k^× be a
group homomorphism and set

    _σ =∑_u∈ Uσ(u)u ∈ kG.

Then Γ_σ:= kG_σ is a left kG-module which is isomorphic
to the induced module _U^G(k_σ), where k_σ denotes the
1-dimensional kU-module corresponding to σ. Note that
_σ^2=|U|_σ. Hence, if |U|1_k≠ 0, then Γ_σ
is a projective kG-module.


We say that σ is non-degenerate if the restriction of σ to
U_s is non-trivial for every s∈ S. Note that this can only occur
if |U|1_k≠ 0. In the case where G=_n(q), the following result is
contained in James <cit.>; see also Szechtman
<cit.>.

  Assume that |U|1_k≠ 0 and σ is
non-degenerate. Then the subspace _σ kG is 1-dimensional
and spanned by _σ n_0. Furthermore, _σ n_w=0
for all w∈ W such that w≠ w_0.


 By the Bruhat decomposition, we can write any g∈ G in
the form g=un_wb where u∈ U, w∈ W and b∈ B. Now note that
b= for all b∈ B and _σ u=σ(u)^-1_σ
for all u∈ U. Thus, _σ kG is spanned by {_σ n_w
 |  w∈ W}. Now let w∈ W be such that w≠ w_0. We shall
show that _σ n_w =0. For this purpose, we use the
factorisation U=U_wU_w_0w where U_w∩ U_w_0w={1}; see
<cit.>. Since w≠ w_0, there exists some s∈ S such
that l(ws)>l(w) and so U_s⊆ U_w_0w; see <cit.>. (Note that U_s is denoted by X_i in
[loc. cit.].) Since n_w^-1∈ n_w^-1H, we obtain

    _σ n_w^-1=_σ n_w^-1=(∑_u_1∈ U_wσ(u_1)u_1)n_w^-1(∑_u_2∈ U_w_0wσ(u_2)
    n_wu_2n_w^-1).

By the definition of U_w_0w, we have n_wU_w_0wn_w^-1⊆ U
and so n_wu_2n_w^-1= for every fixed u_2∈ U_w_0w. Hence,
we obtain

    ∑_u_2∈ U_w_0wσ(u_2) n_wu_2n_w^-1=
    (∑_u_2∈ U_w_0wσ(u_2)).

Finally, since U_s⊆ U_w_0w and the restriction of σ
to U_s is non-trivial, the above sum evaluates to 0. Thus, _σ
n_w^-1=0 for all w≠ w_0. Since w_0=w_0^-1, this also
implies that _σ n_w=0 for all w≠ w_0, as required.

Hence, _σ kG is spanned by _σ n_0. Finally,
by the sharp form of the Bruhat decomposition, every element of Bn_0B
has a unique expression of the form un_0b where u∈ U and b∈ B.
In particular, _σ n_0 ≠ 0 and so _σ kG=1.


  Assume that |U|1_k≠ 0 and σ is
non-degenerate. Then the map φΓ_σ→
kG, γ↦γ n_0, is a non-zero kG-module
homomorphism and every homomorphism Γ_σ→ kG is a scalar multiple of φ.


 The fact that φ, as defined above, is a kG-module
homomorphism is clear; it is non-zero since φ(_σ)=_σ
n_0≠ 0 by Lemma <ref>. Since _σ is a non-zero scalar
multiple of an idempotent (see Remark <ref>), we have _kG
(Γ_σ, kG) ≅_σ kG and this is
1-dimensional by Lemma <ref>.




§ THE SUBMODULE THEOREM
 
We keep the notation of the previous section and assume throughout that
|U|1_k≠ 0. For any group homomorphism σ U →
k^×, we denote by σ^*  U→ k^× the
group homomorphism given by σ^*(u)= σ(u)^-1 for all
u∈ U. Note that, if σ is non-degenerate, then so is σ^*.
We can now state the following version of James's “submodule theorem”
<cit.>, <cit.>.

  Let
σ U→ k^× be non-degenerate and consider the
submodule _σ:=kG_σ n_0 ⊆ kG. Then the
following hold.
=0pt

  (i) If M⊆ kG is any submodule, then either
_σ⊆ M or M⊆_σ^*^⊥.

  (ii) We have _σ⊈_σ^*^⊥
and _σ∩_σ^*^⊥⫋_σ is the
unique maximal submodule of _σ.

  (iii) The kG-module D_σ:=_σ/(_σ∩_σ^*^⊥) is absolutely irreducible and isomorphic
to the contragredient dual of D_σ^*.

  (iv) D_σ occurs with multiplicity 1 as a
composition factor of kG and of _σ.



 Having established Lemma <ref> and Corollary <ref>,
this readily follows from the general results in <cit.>;
the only difference is that James also assumes that _σ=
_σ^*. As our notation and setting are somewhat different from
those in <cit.>, we recall the most important steps of the argument.

(i) Since M⊆ kG, it follows from Lemma <ref> that, for
any m∈ M, there exists some c_m∈ k such that _σ m=c_m
_σ n_0. If there exists some m∈ M with c_m≠ 0, then
_σ n_0= c_m^-1_σ m∈ M and, hence, we have
_σ⊆ M in this case. Now assume that c_m=0 for all
m∈ M; that is, we have _σ M={0}. Let m∈ M and
g∈ G. Using the definition of _σ, _σ^* and the
G-invariance of ⟨ , ⟩, we obtain

    ⟨ m,g_σ^*n_0⟩=⟨ g^-1m,_σ^*
    n_0⟩=⟨_σ(g^-1m),n_0⟩=0,

where the last equality holds since _σ M={0}.
Thus, M ⊆_σ^*^⊥ in this case.

(ii) For u,u'∈ U, we have un_0B=u'n_0B if and only if u=u', by
the sharp form of the Bruhat decomposition. Thus, we obtain

    ⟨_σ n_0,_σ^*n_0⟩=
    ∑_u,u'∈ Uσ(u)σ^*(u')⟨ un_0,u'n_0⟩
    =|U|1_k≠ 0,

which means that _σ⊈_σ^*^⊥ and so _σ∩_σ^*^⊥⫋_σ. Now let M⫋_σ be any maximal
submodule. Then (i) immediately implies that M=_σ∩_σ^*^⊥.

(iii) By (ii), we already know that D_σ is irreducible.
The remaining assertions then follow exactly as in the proof of
<cit.>.

(iv) By construction, D_σ occurs at least once in kG and in
_σ. Let φΓ_σ→ kG be as
in Corollary <ref>. Since Γ_σ is projective and
φ(Γ_σ)=_σ, some indecomposable direct summand
of Γ_σ is a projective cover of D_σ and so the desired
multiplicity of D_σ is at most _kG(Γ_σ,
kG)=1, where the last equality holds by Corollary <ref>.


We now relate the modules _σ, D_σ in
Proposition <ref> to the Steinberg module _k of G, as
defined in <cit.>. Recall that

    _k=kG⊆ kG,      :=
    ∑_w∈ W (-1)^l(w)n_w.

We have _k=|U|; a basis of _k is given by {u |  u∈ U}.

  We have _σ=kG_σ⊆_k. Consequently, D_σ is a composition factor
(with multiplicity 1) of _k.


 By Lemma <ref>, we have the identity

    _σ=∑_w∈ W (-1)^l(w)_σ
    n_w=(-1)^l(w_0)_σ n_0

and so _σ=kG_σ n_0=kG_σ⊆ kG=_k.
The statement about D_σ then follows from Proposition <ref>(iv).


  Gow <cit.> gives an explicit formula
for the restriction of the bilinear form ⟨ , ⟩ kG× kG→ k (see Remark <ref>) to _k. For this
purpose, he first works over  and then rescales to obtain a non-zero
form over k. One can also proceed directly, as follows. We have

    ⟨ u_1,u_2⟩=c_W(u_1^-1u_2)1_k     ,

where c_W(u):=|{w∈ W |  n_w^-1un_w∈ U}| for u∈ U.
Indeed, since ⟨ , ⟩ is G-invariant, it is enough to
show that ⟨,u⟩=c_W(u)1_k for u∈ U. Now, we have

    ⟨,u⟩=∑_w,w'∈ W (-1)^l(w)+l(w')⟨ n_w,
    un_w'⟩=∑_w∈ W⟨ n_w, un_w⟩,

where the second equality holds since n_wB=un_w'B if and only if w=w'.
Furthermore, n_wB=un_wB if and only if n_w^-1un_w∈ B. Since
n_w^-1un_w is a p-element, the latter condition is equivalent to
n_w^-1un_w∈ U. This yields the desired formula.


 
Since |U|1_k≠ 0, the module Γ_σ in Remark <ref>
is projective. Also note that _σ=φ(Γ_σ)
where φ is defined in Corollary <ref>. Using also
Proposition <ref>, we conclude that
_kG(Γ_σ,_k)=_kG(Γ_σ,kG
)=1.
So there is a unique indecomposable direct summand P_ of Γ_σ such that

    _kG(P_,_k)≠ 0.

Being projective indecomposable, P_ has a unique simple quotient
whose Brauer character is denoted by σ_G by Hiss <cit.>.
By Proposition <ref>(iv) (and its proof), we now see that σ_G
is the Brauer character of D_σ.


  It is known that the socle of _k is simple;
see <cit.>. We claim that this simple socle is contained
in _σ∩_σ^*^⊥, unless _k is irreducible.
Indeed, assume that _σ∩_σ^*^⊥={0}.
Then D_σ⊆_k⊆ kG and, hence,
D_σ belongs to the Harish-Chandra series defined by the pair
(∅, k_H) (notation of Hiss <cit.>). By
Remark <ref> and the argument in <cit.>, it then
follows that [G:B]1_k≠ 0. So _k is irreducible by <cit.>.




§ EXAMPLES
 

We keep the setting of the previous section. We also assume now that G
is a true finite group of Lie type, as in <cit.>. Thus,
using the notation in [loc. cit.], we have G=^F where 
is a connected reductive algebraic group  over _p
and F→ is an endomorphism such that some power
of F is a Frobenius map. Then the ingredients of the BN-pair in G
will also be derived from : we have B=^F where  is an
F-stable Borel subgroup of  and H=^F where  is an
F-stable maximal torus contained in ; furthermore, N=N_
()^F and U=^F where  is the unipotent radical of .
In this setting, one can single out a certain class of non-degenerate group
homomorphisms σ U → k^×, as follows.

The commutator subgroup [,] is an F-stable closed connected
normal subgroup of . We define the subgroup U^*:=[,]^F
⊆ U; then [U,U] ⊆ U^*. Furthermore, we shall fix a group homomorphism σ U→ k^× which is a
regular character, that is, we have U^*⊆(σ) and
the restriction of σ to U_s is non-trivial for all s∈ S.
(Such characters always exist; see <cit.> and
<cit.>.) Then the corresponding module Γ_σ=
kG_σ is called a Gelfand–Graev module for G.
Let h ∈ H and σ^h U → k^× be defined by
σ^h(u):=σ(h^-1uh) for u ∈ U. Then σ^h also is a regular character and

    _σ^h=∑_u ∈ Uσ^h(u)u=h_σ h^-1    .

Hence, right multiplication by h^-1 defines an isomorphism between the corresponding Gelfand–Graev modules Γ_σ and Γ_σ^h.

  Let Z(G) be the center of G. Then Z(G)
⊆ H and Z(G)=Z()^F; see <cit.>. Assume now that
Z() is connected. Then there are precisely |H/Z(G)| regular
characters and they are all conjugate under the action of H; see
<cit.>. For any h ∈ H we have

    _σ^hn_0=h_σ h^-1 n_0=h_σ n_0h^-1=
    h_σ n_0

and so _σ=_σ^h. It follows that _σ=
_σ^*=_σ' for all regular characters σ, σ'
of U, and we can denote this submodule simply by _0. By
Proposition <ref>(iii), the simple module D_0:=_0/(∩_0^⊥) is now self-dual. Furthermore, we have

    _0 ≥ |H/Z(G)|.

Indeed, we have _σ n_0∈_0 for all regular characters
σ. Since pairwise distinct group homomorphisms U→ k^×
are linearly independent, the elements _σ n_0 (where σ
runs over all regular characters of U) form a set of |H/Z(G)| linearly
independent elements in _0.


  Let G=_n(q) where q is a prime power.
Then our module _0 is S_λ in James' notation
<cit.>, where λ is the partition of n with
all parts equal to 1. We claim that _0=_k in this case.

Indeed, by <cit.>, _0 is independent of the
field k, as long as (k)≠ p. Since _ is irreducible,
we conclude that _0=_k and the claim follows.
Consequently, by Proposition <ref>(ii), D_0 is the unique simple
quotient of _k. (The facts that _0=_k and that this module
has a unique simple quotient are also shown by Szechtman <cit.>.)
However, D_0 may certainly vary as the field k varies; see
the tables in <cit.>.


See <cit.> for further examples where _0=_k. On the
other hand, Gow <cit.> gives examples (where G=Sp_4(q))
where _k does not have a unique simple quotient, and so _0
⫋_k. Here is a further example.

  Let G=Ree(q^2) be the Ree group
of type ^2G_2, where q is an odd power of √(3). Then G has
a BN-pair of rank 1 and so [G:B]=_k+1. Let k be a
field of characteristic 2. Then kG and _k have socle series
as follows:

    kG      k_G    
       φ_2    
    φ_4   φ_3   φ_5,
       φ_2    
        k_G               _k     φ_2    
    φ_4   φ_3   φ_5.
       φ_2    
        k_G

Here, φ_i (i=1,2,3,4,5) are simple kG-modules and φ_4
is the contragredient dual of φ_5; see Landrock–Michler
<cit.>. By Proposition <ref>, we have
D_0≅φ_3 and _0 is the uniserial submodule with
composition factors k_G, φ_2, φ_3.


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