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ON REPRESENTATIONS OF FINITE GROUPS WITH SPLIT BN-PAIRS
Ahmed A. Khammash
A thesis submitted for the degree of Doctor of Philosophy ~t
the University of Warwick, 1987.
TABLE OF CONTENTS
Acknowledgements Some Standard Notations Introduction Part I. Chapter 1:
BN-pairs
§l.l : Groups with BN-pairs. §1.2 : Groups with split BN-pairs. Chapter 2:
Functor Category
§2.0 : Definitions and preliminaries.
§2.1 : The simple objects of Cov A §2.2 : The functors defined by Green. §2.3 : The kG-module rkG[HJ,kG (k¢) . Chapter 3:
The Steinberg Representation
§3.1 : The Tits complex. §3.2 : Applications to the Steinberg representation. Part II. Chapter 4:
Modular Representations of Finite Groups with split BN-pairs
§4.0 : Preliminaries and basic definitions.
§4.1 : The R-order E(Y) . §4.2 : The simple FG-modules.
Chapter 5:
The R-order S
x
§5.l : The Case when
x is regular.
p-adic Modules
Y(x,J)
Chapter 6:
The
Chapter 7:
The Decomposition Numbers Of The System
(E(KY ) , E(Y ) , E(FY ) X
x
x
§7.l
The irreducible characters of G.
§7.2
The decomposition numbers as multiplicities of ordinary characters.
Y
§7.3
Some contravariant forms on
§7.4
The direct product of split BN-pairs.
§7.5
The case of the general linear group.
Appendix References
X
ACKNOWLEDGEMENTS
lowe a great debt to many people, without whom this work would not have been possible. To my supervisor, Professor J.A. Green, who initiated my interest in this subject and for his patience and guidance during my research time at Warwick University. To Professor R. Carter who has been always ready to give advices and helps whenever I needed them. To my parents, brothers and sisters who have been a great source of inspiration and encouragement during my education time. To Umm - ALQura University, Makkah, Saudi Arabia, for their generous financial support. Finally to Peta McAllister for doing such an elegant typing job.
Some Standard Notations
the cardinality of a set X
xs
T
T\X
X is a subset of T the complement of X in T
If G is a group
Hs G
H is a subgroup of G
Hs G
H is a normal subgroup of G the subgroup of G generated by Hl, ... ,H t s G
If k is a field and M is a kG-module affording the character p
of G the restriction of M to H the restriction of If L is a kH-module
LG
= Ind~L
the
p
to H
(H s G)
kG-module induced from L k
char k
the characteristic of the field
dimkM
the dimension of a k-vector space M
rll.
the Jacobson radical of an algebra In chapter 2 and 3,
A
k is an arbitrary field unless otherwise
,
is stated. Throughout part II,
(K,R,F)
is a p-modular system, where p
>
0
is the characteristic of a finite group G = -(G,B,N,B.,U) with a split BN-pair and k is any field such that char k field for -H = B n N .
t
IHI
and k is a splitting
- i -
INTRODUCTION. Let G = (G,B,N,~,U)
be a finite group with a split BN-pair of
characteristic p and rank t G.
Let (K,R,F)
.
Let
(W,~)
be the Coxeter system of
be a p-modular coefficient system such that K and
F are splitting fields for G and all its subgroup. FG-module FY = FG[U] , where
The permutation
[U] = r u, plays an important role in UEU
the modular representations of G, since it contains every simple FGmodule as a composition factor. This thesis deals with various subjects concerning the FG-module FY. It is divided into two parts (although the division is strictly not necessary). Part I consists of the first three chapters.
The main purpose of Part I is
to investigate some kG-modules (k is a field) arising from some functors defined by Green in [Gl]. Chapter 1 contains the·basic definitions and some structure theorems of finite groups with BN-pairs. Chapter 2 (§2.l)
contains an outline of Auslander theory on the
category Cov A whose objects are functors defined on the category mod A of all finite dimensional left A-modules, where A is a finite dimensional algebra over a field k . For V, X E mod A,
tv,x ' rV,X '
Green [Gl] defined three functors
qv,x : mod
E(V) ~ mod E(X) ,
which connect the representations of the two algebras E(V) E(X)
= EndA(V) and
= EndA(X). The functor qv,x sends every simple left E(V)-module
- ii -
to a simple left E(X)-module qV,X(M) ,
if qV,X(M)
f o.
§2.2
contains the description of these three functors together with some adaptation to the theory of Auslander on the classification of the simple objects in
Cov A.
In this section, we take A = kG,
algebra of a finite group over a field tv,X(M) ,rV,x(M)
and qv,x(M)
the group
k . We consider the E(X)-modules
in the case where M = k$'
a l-dimensional
left E(V)-module affording a l-dimensional character $ of E(V) . In §2.3, we take left kG-module.
V = kG[H],
H ~ G and X = kGkG,
We study the right kG-module
the regular
rkG[H],kG(k$) , when $
is a l-dimensional character of EndkG(kG[H]) . In chapter 3, we take G = BN-pair whose Coxeter system is k-basis kG-module
{Aw; w E W}
to be a finite group with a
(G,B,N,~)
(W,~).
The
k-algebra E(kG[B])
indexed by the elements of W.
rkG[B],kG(k$) , where $ : E(kG[B])
+
X
k
has a
\4e study the right is the multiplicative
character of E(kG[B])
given by $(A) := (_l)t(w) (w E W) where t is w the length function on the elements of W. He show that rkG[B],kG(k$)
is isomorphic to the homology module
Ht-l(~)
(Theorem 3.1.14), where
is the simplicial complex of G defined by Tits [JTJ and t
~
= I~I .
Consequently, we are able to recover some well-known results about the Steinberg representation of finite groups with
BN-pairs (§3.2).
particular, we give an easy proof of the Tits-Solomon theorem on
In Ht-l(~)
,
(their proof involves a less obvious geometrical argument). Part II. Chapter 4 contains an introduction to the modular representations of finite groups with split BN-pairs. E(Y) = EndRG(RG[UJ)
The structure of the R-order
was discussed in §4.1.
In §4.2, we outline the
- iii -
theory of Curtis-Richen on the classification of the simple FG-modules, where G =
is a finite group with a split BN-pair.
(G,B,N,~,U)
The RG-lattice Y = RG[U] has a decomposition Y = L~ Y , where
x€H x
X
tV
tV
H = BnN , H = Hom(H,K) and Y
x
(x
€
is the weight subspace of Y
H)
of weight x (Prop. 4.1.10). In chapter 5, an R-orders
{S, x x
R-order S
x
tV
€
E(Y) . We give an
is introduced, for every x
tV
€
H
The
H} contains all the information about the R-order R-basis for S (x
tV
€
H)
(Prop. 5.0.7) and in §5.l,
x we study the F-algebra FS = F &S , where x x R x
tV
€
H is regular.
We
determine all the simple right FS -modules and its Cartan matrix which
x
turns out to be singular in general. In chapter 6, we give a formula for calculating the characters of the RG-summands of Y = RG[U] , using the results of N. Tinberg [NT2] introduced in §4.2. In chapter 7, we discuss the decomposition matrix D of the Hecke
x
tV
algebra EndKG(KY ) (x € H) , using a recent theorem, due to Green [G2], x which gives an interpretation of the decomposition numbers as multiplicities of ordinary characters of G . In §7.2, we show that the problem of calculating D
x
(x
tV
€
H)
can be
reduced to the case of the Levi subgroup of some parabolic subgroup GJ of G. system
In the end of §7.2, we assume that G = G(q) of finite groups with BN-pairs.
[CIK] and show that the decomposition numbers
~/e
~f
is a member of a
appeal to some results in the Hecke algebra E(KG[B])
can be interpreted as multiplicities of ordinary characters of the Heyl group W of G (Prop. 7.2.31).
- iv -
'V
In §7.3, a notion of a-contravariant forms on Y (x E H) is x introduced, where a: G + G is an anti-automorphism of G satisfying certain axioms.
We show that the set {Y
; x
'V
H \ is closed under the x a-duality and that the simple FG-modules are self a-dual. Consequently, E
we are able to show that the decomposition matrices the Hecke algebras
E(KY) x
and E(KY
woX
D and Dw of x oX
), respectively, are identical,
where Wo is the unique element of W of maximal length. In §7.4, we consider the direct product of two finite groups with split BN-pairs. He prove some results which are relevant to the subject of chapter 7. §7.5 contains an application of the previous results to the case of the general linear group.
- 1 -
CHAPTER 1.
BN-pairs
§l.l Groups with
BN-pairs
Definition 1.1.1
(Tits [JTJ):
A group G is said to have a BN-pair
if there exist subgroups Band N of G such that, (i)
G = ,
B n Nq N .
'" of (ii) The group W= N/BnN is finite and is generated by)set
(iii) For all (iv)
wi
E
Rand w E W, wi B w c BwB u B wiw B , and
w.1
For all
B
w.1 f
B •
The group W= N/BnN is called the Weyl group of the BN-pair, and Write H = BnN,
is the rank.
£
The notation wB, n
E
and let t: N + W be the natural map.
Bw, w E W means
N such that t(n)
= W;
nB,
Bn,
respectively, for any
note that they are well-defined since H ~ N ,
Hs B . The following theorem relates the
(B,B)-double cosets in a group G
with BN-pair to the elements of the Weyl group W. Bruhat theorem 1.1.2
([NBJ, Th.l, p.25)
Let G be a group with a
BN-pair with Weyl group W then: (i)
.
G = U BwB, WEW
(ii)
BwB
=
BwlB implies w =
WI
,
for w,w l
E
W.
o
Matsumoto [HMJ showed that the set R generates W as a coxeter
- 2 -
group;
thus
W has a presentation
W =
1
J
1
is the order of w.w.
+ j)
(i
1 J
Therefore W is isomorphic
to a finite group generated by reflections in t-dimensional euclidean ([NS] pp.25, 91).
space
a set of roots ~
([Re],
has a subset
To such finite reflection group one can associate 2.2); we will denote the set of roots by
11
= {a l , ... ,at}
, whose elements are called the
,
fundamental roots, such that every w.
W can be identified with the
E
reflection in the hyperplane orthogonal to
a. , 1
t
a
E
~
is a linear combination
non-negative or each
,
E
. 1 1=
~
)..
'
a. ,
and such that every
where either each
A.
,
is
is non-positive.
A.
A. s O} 1
~+ ,~
are called the sets of positive and negative roots respectively.
Every root is the image of some fundamental root under some element of W If w E W,
let t(w)
be the minimal number of terms in an expression
of w as a product of generators w1.• w = w.
'1
expression for w, length.
if s
= t(w).
~ ~,
the parabolic subgroups of W.
c
~-, and w~
let W = .. J
generate W as a coxeter group. J
1s
W has a unique element Wo of maximal
We have t(w o ) = I~+I , wo(~+)
For every subset J
is called reduced
• ••• w.
=
1 .
The elements of J
The subgroups W J
J
~ ~,
are called
If we let VJ be the set of linear
- 3 -
combinations of the set
{n
then WJ acts on VJ as i / wi € J} a euclidean reflection group. Let ~J = ~ n VJ ~J is the root system of WJ . The set IT J = IT n V = {n.1 / w.1 E J} fonns the set J of fundamental roots in
~J
Definition 1.1.3
. Let GJ
= BWJB .
A Borel subgroup of G is a subgroup conjugate to B .
A parabolic subgroup of G is a subgroup conjugate to GJ for some J oS R • Theorem 1.1.4 (Tits, [JTJ): J s
are the only subgroups of G
(1)
The subgroups GJ containing B.
(2)
Two different parabolic subgroups which contain a common Borel
~,
subgroup are not conjugate in G. (3)
The normalizer of·a parabolic subgroup is itself.
o Lemma 1.1.5
If G has a BN-pair
(G,B,N,~)
then,for all
J s R ,
GJ has a BN-pair (GJ,B,NJ,J), where NJ is the inverse image of WJ under t. o Theorem 1.1.6 (Curtis, [C1J): (G,B,N,~),
Let G be a finite group with a BN-pair
and let W be the Weyl group of G. Then the map
GJ + WJ (J s~) gives a bijection between the family of parabolic subgroups GJ of G, which contain B, and the family of parabolic subgroups WJ of W. Furthermore there is a well defined bijection WJ w WK+ GJ w GK (w E Wand J,K s~) between the set of (WJ,W K)cosets of W, and the set of (GJ,GK)-cosets of G .
- 4 -
§1.2 Groups with split BN-pairs Definition 1.2.1
A group G is said to have a split BN-pair of rank
and characteristic p,
for some prime p
(i)
G has a BN-pair
(2)
B = UH where U is normal
(G,B,N,~)
>
0,
of rank
R,
if: R,
•
p-subgroup of B, and H is
pi-subgroup of B.
The axiom (3) is called "the saturation axiom".
We write
(G,B,N,~,U)
for the split BN-pair of G. For each w € W we choose
(w)
€
N such that t((w)) = w (i .e.
(w)H = w € W) . Definition 1.2.2 U+
w
For every. w € W define: ww U- = U n U 0
= U n UW ' w
U., = Uw.
,
w.
=Un U1
w. ,
Since H normalizes
U-1. =
U,
U
w
= U0
these definitions are independent of the
choice of the coset representatives ~he
'U.,
(w) .
proof of the following consequences of the axioms of the split
BN-pair can be found either in [C2J or [FRJ. Theorem 1.2.3 ([FRJ, 3.4):
For w € W,
U = U- U+ = U+ U- and U- n U+ = {l} . w w w w w w
o
- 5 -
Definition ([Cl], p.351). lemma 1.2.4
For wi
E:
(See [FR])
(1 )
The coset representative
(2)
(3)
The subgroups in
=
(w. ) 1
can be chosen in
U.H. ,
1
U.1 H.1 U U.1 H.1 (w.1 )U.1 .
G = Gl(n,q),
Example 1.2.5
p
•
If x
(4)
U .> n H . B. ' let H.1 = +
w(ct)
II+ = {ct € ell +
w(ct) € ell+}
W
€
where
o let
4>-}
{U Oct € ell} ct' are defined on which the Weyl group W of G acts in the same
In the next proposition, a collection of subgroups of G
For every ct.1
way that W acts on ell
Proposition 1.2.8 (See [CRIll],
€
II , 1et Uct : = U1 1 0
0
Proposition 69.2):
be a finite group with a split BN-pair of rank
~
Let G =
(G,B,N,~,U)
and characteristic p.
Then (i)
There exists a bijection ct
the set of conjugates {nU 1 0
ct 1 0
+
U1 , ct € II . For all i 0
:
n € N , ct
+ 0
1
ct from the root sys tern ell to € II} , which extends the map U
w € W and ct € ell
,
we
have
(w)u - U ct - w(ct) (i i ) For each w € W we have U-=
w
II U - ct ct€IIw
with uniqueness of expression. A similar statement holds for U+w and In particular,
- 7-
CHAPTER 2.
The Functor Category Throughout Chapter 2,
dimensional
k is a field, and A is any finite
k-algebra.
§2.0 Definitions and preliminaries Let A be a finite dimensional
k-algebra.
By mod A we mean the
category of all finitely generated left A-modules and we denote by Cov A [Fun AJ
the category whose objects are all covariant [contravariantJ
k-linear functors
F: mod A + Mod k, where Mod k denotes the
category of all vector spaces over k. the morphisms a: F, ([RJ p.43).
+
If Fl ' F2
F2 are natural transformations from Fl
If V, X E mod A , we let (X,V)A
E(V) = EndA(V)
= (V,V)A
[( ,V)J , which sends
X mod A to the k-space If F',F
subfunctor of F (written F' s F)
natural in X,
=
. For every A-module V,
is an object of Cov A [Fun AJ.
a k-subspace of F(X)
Cov A [Fun AJ ,
E
E
to F2
HomA(X,V)
and
the functor (V, ) (V,X)A [(X,V)AJ ,
Cov A we say that F'
if for every X € mod A,
and the inclusion map iX: F'(X)
that is if Y € mod A and f
€
(X'Y)A
+
is a
F' (X) F(X)
is
then the
following diagram F(X) ix
F(f) >
t
t
F' (X)
F(Y)
~
F' (f)
iy
F' (Y)
commutes. A functor F
€
Cov A is said to be simple if it is non-zero and
has no proper s ubfunctor.
is
- 8 -
If F' s F
€
COy A we define the guotient functor F/F'
€
COy A
to be the functor gi yen by (F/F' )(X) := F(X)/F'(X) and F/F'
takes any morphism f
for all (X'Y)A'
€
X € mod A , X,Y
€
mod A ,
to the
k-map (F/F' )(f) : F(X)/F' (X) induced by F(f) : F(X)
+
F(Y)/F'(Y)
+
F(Y)
A-submodule of X2 , we write Xl s X2 . A §2.1
The Simple Objects of COy A : ([MAJ, [G1J) In [MAJ, Auslander studied the category Fun A (he named it
Mod mod A) and characterized the simple objects of this category. this section we outline the
~heory
In
of Auslander, using the category COY A
instead of Fun A . Given F
€
COy A and V € mod A,
the k-space F(V) has a
structure of a left E(V)-module by setting: h.x
:=
F(h).x for all
The "evaluation functor"
h
E(V) ,
€
X
€
F(V) .
e : COY A + Mod E(V) , for every V € mod A V
is given by:
for any object F and morphism a:F l The functor eV induces a map:
+
F2 in COY A .
- 9 -
e
~
: {all sub functors F'
V
Definition
([Gl], 3.1):
F}
If M
{all E(V)-submodules of F(V)} •
+
~
F(V) ,
and
X € mod A define:
E(V) bV(M)(X) := {x bV(M)
F(X) I F(g)(x)
€
defines a subfunctor of
s F(V)} E(V)
~1
b : {all V
Proposition 2.1.1
+
F,
€
M,
and
{all
F'
V9 b
V
€
(X, V)}
.
induces a map:
~ F} •
([A], p.28l,
see also [GlJ, §9):
=M
M s F(V). E(V)
Given
F and
V
as above: (i)
eVbV(M)
=F .
(iii) bV(F(V)) (iv)
If Ml Ml If
(v)
~
and M2 are
M2 F'
for all
implies
F(V)
>
Ft V) •
F'(g) FI( X)
Hence,
i.e.
X E
F(X)
x
bV(F'(V))(X) = bVeV(F')(X)
E
=>
F(g)(x)
E
Therefore F'(X) s (bVeV(F'))(X)
F' (V)
for all
for all
9
E
(X,V)
X € mod A .
To complete the proof we need to show that the inclusion map
M,
- 11 -
iX : F' (X) a A-map
+
(bVeV(F') )(X)
(X,V
mod A).
€
is natural in X.
So let e
X + V be
Consider the following diagram:
F{ e)
F(X)
F(V)
>
r
bVe V(F
I )(
bvev(F' ) (X)
i
e)
bVeV(F ' )(V)
>
2
r
fix F' (e) F' (X) Since F'
~
iV
F' (V)
)
F and bVeV(F') = bV(F'(V»
F,
~
it follows that the outer
diagram and diagram 1 commute, therefore diagram 2 also commutes. This completes the proof of (ii). (iii) and
(iv)
follow directly.
(v) To prove (v) assume that F'(V) f F(V) ,
then F'(V)
i
iV
P (f)
F (V) Therefore
(V,f) maps
F'(V)
=
into F'(X) , hence
Since that is true for arbitrary X € mod F'(X)
(V,f)
(V, )(f))
F(X)
>
I
(here
rx
A
(V,f)(lE(V)) (=f)
and f
€
(V,X) ,
€
F'(X)
we then have
= (V,X) , and so F' = (V, ) , which contradicts our hypothesis.
Therefore F' (V)
is a proper left ideal of E(V) .
o
Now we come to a theorem due to M. Auslander which characterizes the simple objects in Cov A . Theorem 2.1.3 (Auslander [MA), 1.6, p.275) (i)
If V € mod A ,then eV and bV induce a bijection: e
{F '
€
det g =
GL(2,q),
l}
,
G has a split BN-pair
0 .
»
,
t
t- 1)
,
(h- 1 -h), h,t ElF; t e: F
H
t- 1)
,
=tC )
,
U
A
€ F; ,
A
€JF
1
(G,B,N,B.,U) ,
A < Fq } ,
q
}
q
}
X
where q is
I'
, and ,
where JF q = GF(q), the finite Galois field with q elements, and x • Fq = Fq\{O}. The Weyl group W= NIH = S2 = = .
(ii) kY A ~ kY~ as
Suppose that x E H.
kY w.>.' ,
f p,
....
and let x E H.
induced bye.>. A( ) k' wi wi'
,
w. E -R and all
kG-modules, for all
A E (x) . A,~ E (x) .
is a
Then
- 67 -
Proof the
(i)
We may assume that
kG-map induced by
W( 1 A.
e A{) A wi' k
•
Let
,
S
Consider
kY . + kY be WA A
sa
E
E{kY,)
•
sa
is
A
induced by e A e A = e A2 A {w.),k W.A {w.),k A (w.),k
,
by 5.0.6
,
"
=e{lu.IA{ )2 k +{L*A 1 )A{)k' A , Wi' XEU. (w.)- h.{x)(w.) wi'
,
, , ,
,2
= IU·IA({w.)2)e, + (L w.A(h.(x))) e A , , A XEU"!"" A (Wi) ,k
since (w.)
,
=
Similarly cl
=
as
2
lu.IA{{W.) )e,A , ,
E(kY
E
, , ,
IU.I w.A((w.)2)
all non-zero in -1 c sa = e and A hence
,
,) W'A k.
E
k,
c
W = w.
If
A,ll E
•
then
II
wiA 1 A
and k
, ,
w.A{{w.)2)
A is a k-isomorphism and
for some
W E W.
be a reduced expression of w.
Let
Then, by (i), we
'm
have a sequence +
kY
W. A
+
'm
of
kG-isomorphisms.
Therefore
kY
II
~ kY
A
as
are
and so we have
kG-map.
= WA
E
,where
A({w,.)2)
Therefore
since it is a
(x)
,
W.A
are non-zero in
,
w.••• w.
','2 ,
cl
and
cle
lu,.1 ,
Since
a.cl-ls = e W.A
kG-isomorphism,
(ii)
by 5.0.13, since
is induced by
by5.0.10
kY
A
kG-modules.
o
H
- 68 -
Since the multiplication relations in 5.0.10 involve and since
lu.1 1
is a power of p ([C2J, 3.3)
the multiplication rule in FS
(hence
IU.I 1
will have a simpler form.
X
lUi I , €
nR) ,
In fact we
have Lemma 5.0.15 (i)
For
Let X A,~
€
~
€
(x),
•
we have
v If w € W, W.1
(i i )
€
R,
and A € (x),
-
v,W
€
W.
then if t(w.w) 1
>
t(w)
t(w.w) 1
the Jacobson radical of FS
X
J1
>
J2
> ••• >
J
~o
>
0 is the radical series of FS
X
Clear from 5.1.3.
(ii) We use the induction on So let 1
X
0, where ui is the simple root corresponds to w. It follows that t(v) ~ n and if A € (x)
,
then eA A(w),F
= e A A(wiv),F = e A A(wi)(V)h',F
= eA Ah',F = A(h')
for some hi
€
H
A{wi){v),F
,
by
eA A(v),F A(w.),F
Hence I n+l S JnJ l and so J~+l = JnJ l = I n+l ' induction and hence the proof of (ii).
5.0.15(ii)
which completes the
(iii)
It is clear from (ii) that J l is nilpotent 2-sided ideal, hence J l s r(FS ) ([CRII], Prop. 5.15). To show that r(FS ) s J l ' x x it is enough to show that FSx/J l is semisimple (see [CRII], Cor. 5.2). It is clear that FS /J l = LlD F.(eA+J l ), and that for all X A€ (x) if
A
= II
if
A
f
A € (x) ,
II
and for all
II €
(x) , wi
€ ~ •
Therefore, putting SA := F.{e A + J l ) , SA is a one-dimensional (hence simple) FS -module affording the character
x
- 73 -
e
~
0
+
e ~ A( w. ) , F
0
+
#
for all
~
for all
~ E
A
(x)
and all
W. 1
1
A,~
It is clear that if
F\/J l J
l
(x)
E
is semisimple right
then
1jJA
# 1jJ~,
F\ -module and so
if
A
!.(FS ) x
#
~
~.
Jl
R
E
Hence Therefore
= reFS ) .
-
X
(iv)
Jr/J +l r
It is enough to show that
F\-module or equivalently this follows from (iii).
Jrr(FSx)
(= JrJ
is semisimple right
l ' by (iii)) = J r +l
This completes the proof of 4.3.20. {SA' A
set of simple right
But
E
(x)}
o
fonn a full
FS -modules.
x
As a corollary to the proof of 5.1.5{ii) we have Lemma 5.1.6
1 ~ r ~ "0'
For every
J~
is generated as right
FSX-module
by the set
WE
{e A A{w),F / A E (x), For every
A
E
(x),
let
projective indecomposable right
It is clear that
P).
= L~ F.e~A( WEW
f\
dim
P F A
) F' W,
= Iwi
P = e A
W with
A
FS
.
x
i{W) = r} From 5.1.4,
FS -module for all
A E (x) ,
for all
in fact
X
By 5.1 .5(iv),
A
E
(x);
o PA
is a
and
PA has a Loewy series ([PLJ, p.27)
- 74 -
5.1.7 where, for all Xr =
0
~
r
~
')til' ,
PAJ~ = w~:
F.e A A{w),F
R.{w)~r
Fix A E (X),
o~
r
~"~-l
and consider the semisimple quotient Xr/Xr+l
, in the Loewy series 5.1.7 of
PA .
xr IX r+l = Em WEW
R.{w)=r If
~ E
(X), and wi
E ~,
then by 5.0.5
if
~
t-
if
~
= wI..
wI.. ,
and (eAA{W),F + Xr+l)e~A{wi),F = eAA(w),Fe~A{wi),F + Xr+l if
~
t-
if
~
= wI.. and R.{wiw) = R.(w) + 1 .
wI.. or R.(w;w)
But eA A{wi)(w),F is a scalar multiple of eA A{wiw),F (since R.{w) = r),
therefore
E
F R.{w) + 1
Xr+l
- 75 -
From (*) and (**),
it follows that F.(e1A(w),F + Xr +1 ) is onedimensional right FS -modu1e which affords the character ~WA ' x 'V hence F.(eAA(w),F + Xr +1) = SWA Summarizing the above we have Proposition 5.1.8
Let x
~
Let PA be the projective indecomposable right FS -module whose Loewy series €
be regular, and let A € (x). X
is given by 5.1.7. (i)
Then
0
~
r
{e A A(w),F
I
w € W with t(w) = r}
For all
~
t,
Xr is generated as an FSx-module by the set
(i i )
Xr IX r+l '.!'- E~ S WA ' WEW t(w)=r
0 ~ r ~ t-l .
(iii)
w € W, SWA ' .appears as a composition factor of PA with multiplicity 1 .
for a 11
For all
Follows from the fact that Xr = PAJ~ and using 5.1.6.
Proof (i)
Clear since Xr/Xr+1 = w~:
(ii)
F.(e A A(w),F + Xr +1) , and
t(w)=r 'V
F.(eAA(w),F + Xr +1 ) = SWA (i>ii) Follows from (ii) and the fact that dim F PA = Remark
It follows from 5.1.8
F~a1gebra
FS
X
1 1 C
=
(
In particular, C is singular unless
*
o
( iii) th a t th e Cartan matrix C of the
has the form
*
IWI
*
11
:tXIWI
Iwi = 1 *
*
- 76 -
S is an Let X E (x),
element e(x)
F-~~algebra
A
(w), F
w E W,
with identity
X
and consider the projective indecomposable
right FSx-module Px as right S-module. WE W
Therefore, for all
of FS
For all
~
if ~
f wx
if
= wX
~
E (x) , and all
F.eXA(w),F is a one-dimensional right
S-module affording the character e
~
e wx
+
0
+
1
if
~
f wX ,
if we let MT = Li F.e x A(w) F wET ' then MT is a right S-submodule of Px ' Conversely, if M is an
Hence for every subset T
S-submodule of
PA
some w E W,
,
of W,
then for all
in which case
Me~
~
E (x) ,
= F.eA A(w),F
Me~ ~
f0
M,
~
= wx for
hence
M=
L F.eXA(w) F = MT for some subset T of W. It is clear from wET ' the above argument that, if Tl , T2 are subsets of W, then Therefore we have Tl s T2 MT is S-submodule of MT
1
Proposition 5.1.9
2
The map T + MT gives a lattice isomorphism between
the lattice of subsets of Wand the lattice of S-submodules of PA • 0 Now, for a given subset T of W, we would like to know under which condition the S-submodule MT of Px will be an FSx-submodule.
- 77 -
We need the following definition. Definition 5.1.10 If v,w
(i )
€
W,
write v
such that v-1 (a i ) (ii)
(Bruhat ordering):
E ~
.w
if w = w.v for some w.1 1
~
+
i.e. such that t(w.v) 1
Say v
~
w (v,w
sequence v = YO' v1, ... , vn = w all 0 ~ i ~ n-1 .
€
W)
€
R
t(v)
>
if there exists a
such that vi ~ vi+1
for
Now we return to the S-submodule MT of PA , where T s W. 5.1.3, if w € W, then for all Wi € R and all II € (x) , if
II
= WA and t(wiw)
By
>
t(w)
5.1.11
if either II f WA or t(wiw) < t(w) Definition V
€
A subset T of W is said to be "good" if WET implies
T for all
v E W with w ~ v . For a given T s W,
Lemma 5.1.12
the S-submodule MT of PA is an FS -submodule if and only if Tis "good". X
Proof
Suppose that MT = w!T F.e A A(w)
Since FS {ell
X
A(w.),F
is generated as an
I
II
€
is FS -submodule of PA . x
F-algebra by
(x) , Wi E R},
it follows that if wET then
1
II €
In particular, if
II
= WA and t(wiw)
>
(x) and all t(w) ,
w·1
E
R•
-
then by 5.1.11
- 78 -
eA A{w),F el.l A(w. ) ,F = eA A(w.) () w ,F 1 1
€
MT .
But since eA A{w.){w),F is a scalar multiple of eA A{ wiw ) ,F 1 it follows that eA A{w.w),F
€
1
~(wiw) > ~(w).
Hence wiw
MT , for every wi T for all
€
wi
€
hence T is "good". that wiw
€
T,
€
R
R with
or equivalently T contains all the elements v Consequently T contains all the elements v
€
€
'
with ~(wiw) > ~(w)
W with w ~ v .
W with w ~ v,
and
Conversely if T is "good" then w € T implies
for all
wi
€
R with
~(wiw) > ~(w),
and reversing
the above argument, we have W. 1
Hence MT is an FSx-submodule of PA
€
R •
o
,
- 79 -
CHAPTER 6.
The p-adic Modules Y(x,J)
We saw in 4.2.2(i)
that if x
E
~
then
FY = e-(FY) = E~ FY(x,J), x x J~P (x)
6.0.1
where the FG-modules FY(x,J) are pairwise non-isomorphic indecomposable. Let 6.0.2
ex
=
E
J~P (x)
e(x,J)
be the orthogonal idempotent decomposition of e- in E(FY) which
x
corresponds to the decomposition 6.0.1. 6.0.3
x
It follows from 4.1.9. that
e(x,J.)
lE(FY) = x~ft J~P (x)
is an orthogonal idempotent decomposition of lE(FY) E(Y)
in E(FY).
Since
is complete with respect to the TIE(Y)-adic topology, we can lift
the equations 6.0.2 and 6.0.3 to get orthogonal idempotent decompositions 6.0.4
e x
=
o
E
J~P (x)
e (x,J) ,
o
6.0.5
e (x,J)
of ex and lE(Y) , respectively, in E(Y) . The idempotent eO(x,J) in E(Y) 6.0.6
is such that eo (x,J)
= e(x,J)
- 80 -
for all admissible G-pairs Definition
(x,J).
We write eo(x,J)(Y) = Y(x,J) .
Since eO(x,J) G-pairs
is primitive idempotent in E(Y) , for all admissible Y(x,J) = eo (x,J)(Y) is an indecomposable RG-lattice.
(x,J) ,
It is clear that Y(x,J) = FY(x,J),
and that the Y(x,J)
are pairwise
non-isomorphic RG-lattices since the FY(x,J) are pairwise non-isomorphic. It follows from 6.0.4 and 6.0.5 that 6.0.7 . 6.0.8
Y = e (Y) = La Y(x,J) , J~P(x) x x Y=
La
xEft
e (Y)
x
= L~
and
Y(x,J)
XEH
J~P(x)
In this chapter we use the- results of N. Tinberg, introduced in §4.2, to calculate the characters of the RG-lattices Y(x,J) . Similar calculations were done
independently by P. Sin [PSJ.
Replacing G by GJ , J lE(FY ) = x;~ J
~ ~,
in 6.0.3, we get (see §4.2)
eJ(x,S).
S~PJ(x)
Therefore we have a decomposition of the RGJ-lattice YJ ' similar to 6.0.8; namely YJ = L~ XEH
YJ(x,S)
S~PJ(x)
~ G Since YJ = RU J '
where RU
is the trivial
RU-lattice, and since
- 81 -
G G (RU J )
= R~ = Y
,
it follows from 6.0.8
G th at YJ (x , S) , as a component(*)
G
of RU'
components, each of the form Y(x,L),
Since the inducing functor
Ind~
(using Krull-Schmidt theorem) is isomorphic to sum of
say
commutes with the reduction mod nR , J
we have 6.0.9
. Comparing 6.0.9 with 4.2.3 we are able to calculate the mL gives Lemma 6.0.10
Let J
~
R and let (x,S)
this
be an admissible GJ-pair.
Then,
o Putting J = P(x)
in 6.0.10 we have
6.0.11 for every admissible G-pair
(x,S) . Therefore, in order to determine
the character of the RG-lattice Y(x,S) , we only need to calculate the character of Yp(x)(x,S) . (*) The components of an RG-1attice M are the direct RG-summands of M .
- 82 -
Hence we assume that x
€
~
is such that P(x)
= R.
Using
6.0.10 we have 6.0.12
Lemma 6.0. 13
For any J 5!,
YJ(x,J) is aone-dimensional
RGJ-lattice.
The lemma follows from the fact that YJ(x,J) = FYJ(x,J)
Proof
FGJ-module
dimensional
([NT2], lemma 4.10).
The one-dimensional an element mJ(x,J)
. GJ-action on mJ(x,J) umJ(x,J) 6.0.14
(see [NT2],
(J 5!)
is generated by
proof of 2.10), with the
given by
= mJ(x,J)
(wi)mJ(x,J) hmJ(x,J)
o
FGJ-module FYJ(x,J)
FYJ(x,J)
€
is one-
= mJ(x,J)
= x(h)mJ(x,J)
for all
u
for all
w.
for all
h
U,
€
, €
€
J
, and
H.
Let px,J be the one-dimensional character of GJ afforded by FY J (x ,J) . By 6.0.14 px,J(g) € (FX)c for all g € GJ , where c is the exponent of H (see 4.1.4). Therefore Px, J can be lifted to a K-character po J of GJ by means of the map r of 4.1.4; i.e. x, If a is a k-character of p~,J(g) := r(px,J(g)) , for all G and J 5
~,
denote by
al GJ
the restriction-of
a
to the parabolic
subgroup GJ . Lemma 6.0.15 o
x,_R
P
•
(i)
The
RGJ-lattice Vex,!)
affords the character
- 83 -
(ii) Proof
P~,~
= pox, J
pox,_RIG J
(i)
Let T be the
RU-lattice.
for all
J sR.
RG-lattice which affords the character
p~,~lu
By 6.0.14,
.
'
=
lU'
and so Tlu
=
RU '
Since U is a Sylow p-subgroup of G,
([CRI], Cor. 63.8).
Hence T is a component of
Therefore T ~ Y(xl,S)
the trivial T is U-projective
(Tlu)G = R~ ~ Y .
for some admissible G-pair
(Xl,S).
the FG-module FT = F 9 T should afford the character p R R x' pox,_R is the lift of px,_R). Hence FT = FY(x',S) = FY(x,~) (x ,S) I
But then (since and so
= (x,~) and T = Y(x,~) (ii) Clear from 6.0.14.
For L s Y(x,L).
p
A =
have Al
1et T\ be the character afforded by the
By 6.0.12,
6.0.16
Let
~,
RG-l attice
we have
o G J = E nL x, J~LsR
for all
J
~
R .
X
(po R)-l : G + K x,_
po GJ x,J
= lG J
'
Since po RI = po J (6.0.15(ii)), we x,_ G x, J and so, by Frobenius reciprocity, we have •
6.0.17
From the equations 6.0.16 and 6.0.17, 6.0.18
G
lG
J where
=
E J~L~B
n1 L '
for all
J
~
we have R,
Solving equation 6.0.18 for nl,J
([NB], Exercise 25,
- 84 -
p.44-45), we get 6.0.19
Therefore we have Proposition 6.0.20 and let nJ' Y(x,J).
. Proof
J
~
Suppose X € ~
B,
= Hom(H,Kx )
=B
is such that P(x)
be the character afforded by the
RG-lattice
Then
Clear from 6.0.19,
since nl,J = AnJ and A
o In particular, if
o
~
= Px,_R' StG and
denotes the empty subset of R then
(see 3.2.4),
- 85 -
CHAPTER 7.
The Decomposition Numbers Of The System (E(KY ), E(Y ), E(FY )) x- x- x
Let x € ~ and let J s ~.
Consider the R-algebra E(Yx,J) = End RG (Y J)' Where Y J = eJ(y J ), and eJ = __1__ L x(h- l )A J J x, x, x IHI h€H h J Since ex is an idempotent in E(Y J ) Yx,J is a direct RGJ-summand of YJ . Since YJ is p-endostable RGJ-lattice, it follows (see [G2J, Lemma 2.4) that Yx,J is also a p-endostable RGJ-lattice. then k & E(Y J) ~ E(kY J) , and so we may
Therefore if k € {K,F}
R
x,
x,
regard E(Y x, J) as an R-order in E(KY x, J) . Let E J:= e J E(YJ)e J , x, x x J J KE x, J := ex E(KYJ)e x , and FE J = e~ E(FYJ)e~ (note that if x, x x IV
If J = ~, we write Y = Y then kE x, J = E(kY x, J)) x,~ x = ElY and if k € {K,F} , we write kY = kY and kE = kE
k € {K,F} E
x,~
X ,~x
I'
We saw in §4.2 that if X € ~, (&
FY = L
x
SsP (x)
X ,~
X
then
FY(x,S),
where FY(x,S) are mutually non-isomorphic indecomposable FG-modules. Consequently the simple modules for the F-algebra FE
(which is x are all one-dimensional indexed by the set
isomorphic to E(FY)) x ((x,S) I ,S s P(x)} (see [CRIIJ, Prop. (6.3), p.120). We denote by
S(x,S) the one-dimensional right FEx-module which corresponds to the pair
(x,S) . In Chapter 7 we consider the decomposition numbers of the system
(KE , E , FE) X
x
x
using a recent theorem of Green ([G2J, Thm. 4.2), which
- 86 -
relates the decomposition numbers of the system
(KE, E , FE) X
the multiplicities of the simple components of KY
X
X
to
. The next section
X
will contain a parametrization of the simple KG-components of KY
§7.l
X
The irreducible characters of G Let G = (G,B,N,B.,U)
be a finite group with a split BN-pair of
characteristic p. The Levi decomposition
let wJ denote (w ) J the unique element of W maximal length. Let U = U n U and of J J LJ = . UJ is a normal p-subgroup of the standard . parabolic subgroup GJ and GJ has a decomposition GJ = LJU J as a semidirect product of LJ and UJ . Moreover B = HU- and NJ form J wJ ( [RC J,
§
2.6 )
For each J
a split BN-pair of LJ of rank IJ I , where NJ of W J under t. The above decomposition of GJ decomposition and LJ is called Levi subgroup of
~
B.,
is the inverse image is called the Levi GJ .
Let chKG J denote the set of a11 cha racters of GJ over K . 'V
LJ = GJ/U J , each character e of LJ ' over.K, gives a character e € chKG J having UJ in its kernel, in fact eG is given GJ J by Si nce
eG (R.u) .- e(R.), J
for all
let eG GJ
u
€
UJ ,
R. €
LJ
be the induced character
- 87 -
Definitions (i)
(See [RC], §9.1)
X
€
chKG is called cuspidal if for all
e
€
chKL J G J
component of e~ (i i )
= 0,
i.e.
J ~ ~
X does not appear as a
. J
If J l and J 2 are subsets of B. ' we say that J l and J 2 are associated if W(1T J ) = 1T J for some w € W. 2 1
(i i i ) For each J
~
-R let
fl
J = {w € WI W(1T J ) = 1T J }
It follows (see for example [RC], Prop. 9.2.2) then
and all
(W)L J = LJ ' Therefore the group
fl
follows: ifx€ chKL J and We: fl J then Wx
that if W€
fl
J
acts on the set chKL J as
J e:
chKL J is given by
The Harish-Chandra theory for dividing the irreducible characters of G into classes suggests that one can find every irreducible character e of G as a component of x~
for some J ~ R and some cuspidal J
The ,following summarizes Harish-Chandra theory. Theorem 7.1.1 (t)
([RC], Theorem 9.2.3)
Each irreducible character e of G appears as a component of x~
for some cuspidal character X of some Levi subgroup J
L
J
of G .
- 88 -
(ii)
Take one J E R for each class of associated subsets of R and one cuspida1 character x of LJ in each nJ-orbit of cuspi da 1 characters. If we take all irreducible components of G
xG
J
'
In [HLJ,
then we get each irreducible character of G just once.
o Howlett and Lehrer have given a method of parametrizing
the irreducible components of x~
, where x is an irreducible character J
of LJ , by studying the endomorphism algebra of a KG-module affording xGG The K-basis elements of this endomorphism algebra are indexed J
nJ I Wx = x} of W. Using some specializations of a certain generic Hecke algebra, Howlett and Lehrer showed
by the subgroup Wx,J
:=
{w
€
that the irreducible characters of G which appear in x~
are in 1-1 J
correspondence with the irreducible characters of the group algebra (KWx, J) p
twisted by some cocyc1e
p
(see [HLJ, Corollary 5.5).
However, for our purpose we will take J of
~,
of G. If x
€
=~
the empty subset
hence LJ = L~ = H, and G~ = B, the standard Borel subgroup It is clear that all irreducible characters of Hare cuspida1.
~,
then x~ is the character of G afforded by the KG-modu1 e
(= ex(KY)). Recall that the K-algebra KE = e E(KY)e x X x is isox , morphi c to E(KY ) and has K-basis {ex A(w),K ' w € W} , where x x W x = Wx,~ = {w € W I Wx = x} ( no te th a t n~ = W) .
KY
Remark 7.1.2 of W.
The group W x, J is not, in general, a reflection subgroup
However, the theory of Howlett and Lehrer [LHJ shows that W x, J
has a decomposition W x, J M
x,J
= MX, JMx,' J
as a semi direct product, where
is a reflection normal subgroup of Wx, J.
- 89 -
§7.2 The Decomposition Numbers As Multiplicities Of Ordinary Characters For X € ~,
let I be an index set for the complete set of
irreducible characters of the twisted group algebra by Howlett-Lehrer theory,
(KW) X
Then,
Jl
I is also an index set for the set
{e. , 1
i
€
l}
of irreducible characters of G which appear in x~. For each
i
be an RG-lattice such that KX i is a KG-module which affords the character e.1 . Then KX.1 is a simple component of KY
I,
€
X
,
let Xi
for all
i
€
I
The
K-space
(K\, KXi)KG ' i
€
I ,
has a natural structure of a right E(KY )-module, hence a right KE -module
x
X
(since KEx ~ E(KY ))'
In fact
X . (see [CRII], Prop. 6.3, p.120),
(KY ' KXi)KG is a simple KEx-module x for all i € I, and the set
is a complete set of simple right KE -modules.
For each
X
an R-fonn of the KG-module Consider the R-lattice E(YX)-lattice,
Similarly Y ,
X
€
i
€
I,
X.1
is
I
€
Xi
We identify Xi with an
with lK Q x
€
can be identified with an R-submodule of KY
KX i . Con-
X
(Yx,Xi)RG can be regarded as a subset of
(KYx,KXi)KG by identifying f fl.
(Yx,Xi)RG'
by identifying x
sequently the R-lattice
€
KX i .
hence it is a right Ex-lattice.
R-submodule of KXi
i
€
(Yx,Xi)RG with the unique
(KYx,KX i )KG which coincide with f on Y
X
K-map
Since
K : (Yx,X i )RG ~ (KYx,KX i )KG ' . we may then regard the right EX-lattice
- 90 -
(Y ,X.) = F 6a (Y ,X.) = {f, f X 'RG R X 'RG (Note that in general that the pair YX,X., and g
€
E(Y ) , X
€
(Y ,X,,)} X RG
(Y ,X.) 1 (Fy' ,FX')FG' since we are not assuming X , RG x' is p-stab1e.) It is clear that if f € (Y X,X.) ,
then fg
= f g,
therefore
(Yx,Xi)RG is a right
(since F 6a E{Y ) ~ E{FY )). R X X
E{FY )-modu1e X
Hence
(Y ,X')RG is a X ,
right FE -module, which is a p-modu1ar version of the simple right X
KE X-module
(KY X,KX')KG . ,
Definition
(R. Brauer, see [CRII], p.413)
all
i
For every S
~
P(x) , and
I , let di, (x,S ) denote the multiplicity with which the simple right FE -modu1e S(x,S) appe~rs as a composition factor of (Yx,X;) x The numbers d. ( S) are called the p-decomposition numbers of the €
" x,
system (KE, E , FE ) . X
X
X
Let S ~ P{x) and let P(x,S) We may take P(x,S) given ;n 6.0.2.
= e(x,S)FE X
Then, for all
be a projective cover of S{x,S) .
, where e(x,S) i
E
is the primitive idempotent
I, we have (see [CRII], Exercise 10{a),
p. 71 )
7.2.2
di, (x,S ) = dim F (P{x,S), (Yx,X,'))FG
On the other hand, let eO{x,S)
be the lift of e{x,S)
can be regarded as an idempotent in KEx
.
in E . eo (x,S) X
- 91 -
Since eO(x,S) (Y ) = Y(x,S) by definition, we have x eO(x,S)(KY ) = KY(x,S), and since KY(x,S) is a KG-submodule of x KY , we then have x KY (X, S) =
'('@ t..
id
*
d. ( S) KX. , 1,x, 1
*
d.1, ( X, S)
€
71.>_0
•
A more general setting of the following theorem was given by Green [G2]. Theorem 7.2.3 (Green [G2] Theorem 4.2)
If x
€
~,
then
*
di,(x,S) = di,(x,S) for all
i
Remark
The above theorem provides a formula for the multiplicities of
€
I and all
S sP(x)
the simple components of KY
X
o
, X € ~,
decompos iti on rna tri x of the sys tern
provided one knows the
(KE, E , FE ). X
X
X
In
fact if
(X,S)
0
X
.-
i
di,(x,S)
I I
,i
€
I and S s P(x)
I
is the decomposition matrix of the system 7.2.3,
i
[KX.
1
KY ] X
=
[KX.
1
E@
(KE , E , FE ), X X X
KY(x,S)J
SsP(x)
= S~P(x) di,(x,S)
then by
- 92 -
for all KXi
i e: I, where
as a component of KY
*
*
I
[KX i
KY ] denotes the multiplicity of x
x
*
*
*
G
Now let J s
be such that Wx
~
S
WJ .
Let
~
= xB J and let
< '>G denote the usual scalar product on chKG (see for example
[CRII], p.210). Lemma 7.2.4
Proof
By Frobenius reciprocity formula and Mackey decomposition (see
[CRII], p.237), we have
=
1:
we:W
Since H s wB n B, Wx
= x;
i.e.
wBnB
for all
unless w € W X
w e: W ,
But since Wx
S
= 0 unless wBnB WJ , we have
o
- 93 -
Suppose that
~ = E m.~.
i
, where m,.
= j,
~,.
and
is an
"
irreducible character of GJ i
7L 0'
E:
for all
i ,
with
~. =~.
,
J
only if
then we have
Co ro 11 ary 7. 2. 5
,
~~ is irreducible character of G for all
~~, = ~~J only if i = j Proof
G = E m.m. ;,j , J
G G
G i
;
'
"
+ E m.m. i#j' J
G G
G , J
=0
It follows from the orthogonality relations (see [CRII],9.23)
that ~~, ;s irreducible for all
i,
It follows from 7.2.4 and 7.2.5
and that ~~, # ~~J
if i # j
, ~~,
that the map s.
+
o gives a
1-1 correspondence between the set of irreducible characters of GJ which G appear in xB J and the set of irreducible characters of G which appear in X~.
Moreover, the corresponding characters under this correspondence
appear with the same multiplicity.
So we may assume that, for every i
€
I ,
- 94 -
, ,
~~ = a. and hence we have
, ,
m.
L
id
= .,.~G XG B
mi
~.
€
= ~ m ~G - ~ m a t..
iEl
..,..
'
-
t..
,
and
7l>O'
,.
,..
id
Reca 11 tha t the KG J -modu1e KY J = eJ(YJ) = KGJ[UJS affords x, x x GJ the character xB . For every i € I , let X. , J be an RGJ-lattice
,
, , ,
, ,
such that KX. , J is a simple KG J -modu1e which affords the character ~ . Si nce ~.G = a. (i € I) and since a. is afforded by the KG-module KX. ,
,
G it follows that KX. , J
,
'V
=
,
KX. , for all
i
€
I .
We would like to compare the decomposition numbers of the two systems (KE , E , FE ) and (KE J' E J' FE J) , where J E R with W x ~ WJ . X, X,X, X X X Let 0 0 be the decomposition numbers of the systems X ' x,J (KE , E , FE ) , (KE J' E J' FE J) , respectively. Since the X, X, X, X X X number of irreducible characters of G which appear as components of
X~ is equal to the number of irreducible characters of GJ which appear GJ
as components of XB namely
III .
P(X) ~ J',
' oX and 0x, J have the same number of rows,
Moreover, since Wp(X) ~ Wx
(see 4.2.1),
it follows that
and so the simple FE - and FE J-modu1es are indexed by all
subsets of P(x) . Therefore 0
X
col umns, namely
x,
X
and 0 J have the same number of X,
Hence the decomposition matrices
0
X
and
ox,J have the same size. has
R-basis
{e~Aiw) , w € WX}
defines an injective R-a1gebra map
- 95 -
a : Ex, J
Ex
+
(see [NT2J,
p.511).
But since
it follows that a is an R-isomorphism.
rank R Ex,J = rank R Ex = IWxl Hence we have: Lemma 7.2.6 R-orders
If x
~
x,
x
~
Y = X
7.2.8
and J ~ ~ is such that Wx ~ WJ ' are isomorphic.
W J and Wp(x) ~ W}'( Hence, from 6.0.7, we have
J.
7.2.7
~,
E J and E
Since Wx P(x)
€
Y J x,
Q}
E
S~P(x)
=
Y(x,S),
Q}
E
S~P(x)
(see 4.2.1),
then the
o
it follows that
and
YJ(x,S)
By 6.0.10, we also have YJ(x,S.) G 'V= Y(x,S) , for all
~
S
P(x) .
Let 7.2.9
e
=
X
E
S~P(x)
eo (x,S)
, and
7.2.10 be the orthogonal primitive idempotent decomposition of e
x
in Ex and Ex, J 7.2.7 and 7.2.8.
x
respectively, which correspond to the decompositions Since a(e J ) = e , we may arrange the decompositions X
x
7.2.9 and 7.2.10 so that a(e3(x,S)) Since e3(x,s)
J and e
= eO(x,S)
for all
S ~ P(x)
is an R-combination of the elements {e~Atw) , w € W x}
and since eJAJ( )([UJS ) = S [U(w}UJ = e A(w)([UJS ) , for all w € W , X w x X X X X it follows that
- 96 -
= eO(x,S) ([UJe) x
eOJ(x,s) ([uJe )
x
S s P(x) •
S s P(x), we have
Therefore for every KY(X,S)
for all
= eO(x,S)(KY ) x
= eO(x,S)(KG[UJe
x
)
= KG eO(x,S)([UJe ) x = KG e3(x,S)([UJe x ) tV
KYJ(x,S)
=
G
(see [CRII], Prop. 1l.2l).
J * denote the multiplicity of di,(x,S) . composition factor of KYJ(x,S) •
For
i
E
I ,
let
Lemma 7.2.11
Let x J
*
di,(x,S) Proof
d1, . (x,* S)
and
J
be as in 7.2.6.
*
for all
= di,(x,S)
= [KX.1 I
KX.1, J
Then
S s P(x) .
KY (x , S ) J
J * = d.1, ( X, S)
Since and 7.2.11
0
x
and
0 J
x,
0 are of the same size, it follows from 7.2.3
that, under a suitable arrangement, the mat;.rices: S J S
Dx "; (- -
as a
-«~,S)
..
and
ox,J
= i
J
•.• d. ( 1,
*
X' S)
- 97 -
are identical.
Therefore we may replace the group G in theorem
7.2.3 by GJ where J is any subset of
with Wx
~
S
WJ .
Recall that LJ = , . Therefore any x € ~ can be
Now let J be any subset of R. where BJ = U-w H, J
and U-w J
BJ
9
extended to a character xB
of BJ . J
For
X
€
ft, let yJX := RL}U-w JI3 X , KyJ:= KL}U- JI3 and X w X J
FyJ := FL [U- JI3- . X J wJ X
If X € modoRL
denote X regarded
J
acting trivially on X.
as an RGJ-lattice by letting UJ Proposition 7.2.12
J
J ~ R and all
For all as
X
€
ft we have:
RG J - lattices.
Proof
We have GJ = LJU J and LJ S B S GJ . Hence GJ = LJB, and so any left coset of B in GJ has the form tB for some t € LJ . -1 If t l ,t 2 € LJ then tlB = t2B t2 tl € B n LJ = BJ tlB J = t 2BJ . Therefore there is a bijection between the left B-cosets of GJ and the left BJ-cosets of LJ . In particular rank R \,J = /G J : B/ = J Let ~ : Y J BJ / = rank R yx x, which sends t[UJI3 to H U~ J13 (t
/L J
X
X
J
+
J YX be the bijective R-map
€
LJ)'
If u
then utCUJI3 = tt- l utCUJI3 X
X
= u- l utl3 [UJ X
= x(t-lut) t[UJI3 = t[UJI3
X
X
since since
€
UJ'
and t
€
LJ
- 98 -
Therefore UJ acts trivially on Y . Consequently the map x,J ~ : Yx,J + (Y~}GJ is an RGJ-map, for if 9 = £I U E GJ (£1 E LJ,u E UJ ) , then ~(gt[UJS
) =
X
~(£lut[UJS
)
X
,"
= £1 ~(HUJS ) X
= tlu~(HUJS ) X
= g~(t[UJS ) X
for all
t E LJ .
o
J
-
Y = RLJ[UwJJ , J
(where Y = RGJ[UJ and J R-bases indexed by the elements of NJ .
E(Y J } and E(yJ)
The R-lattices ~~)
have
In fact E(Y J ) = E~ R.A~ and E(yJ) = E~ R.A~J nEN J nEN J J and A'n (n E NJ ) are given by
A~([U,J) .- [UnUJ
and
Let X E ~ and let n by'4.1.14, E(Yx,J).
€
NJ
be such that ten} = w € Wx
the restriction of AJn to Yx, J
Similarly, the restriction of A~J
element of E(yJ}. X
,where A~
. Then
induces an element of to
Y~ induces an
It is clear that the endomorphism algebra
- 99 -
E(yJ) (= End RL (yJ)) X J x Consider the diagram Y
>
x,J
A~
7.2.13
1 y
>
x,J
where
~
is the RGJ-map defined in the proof of 7.2.12.
Lemma 7.2.14 Proof
The diagram of 7.2.13 is commutative.
Since the maps of the diagram 7.2.13 are all
RGJ-maps, it is
enough to show that ~AJ([UJB) = A,J~([UJB) or equivalently X
n
n
X
The last statement will follow if we prove that U- n U- -- (U-W )-w W W
G
J
J
u:
From 1.2.8(ii), we have
J
- 100 -
=
o.
be a member of a system of
(W,.!!,) , where q
Let B(q)
is
denote a standard Borel subgroup
of G(q) . Define, in the group algebra KG(q) , the idempotent b(q) = IB(q)l- l l: b and the left ideal V(q) = KG(q).b(q) . V(q) b€B(q) a left KG(q)-module affording the character l~~~~ Definition 7.2.26
(Iwahori [IJ):
a subalgebra of KG(q)
The Hecke algebra HK(q)
is
is defined as
by
HK(q) = b(q) KG(q) b(q) Definition 7.2.27
For all
w € W,
define
,
The Hecke algebra HK(q)
is isomorphic to the opposite ring
(EndKG(q){KG{q).b(q)))oP, where EndKG(q)(KG(q).b(q)) alg~bra
of left operators on KG{q).b{q)
is viewed as an
(see [CRIIJ, pp.28l-282). The
simple HK{q)-modules are in l-l-correspondence ItJith the set of irreducible characters of G(q) which appear in l~~~~
([CRIIJ, Thm. 11.25(ii)).
- 107 -
The structure of the Hecke algebra HK(q)
has been determined by
1wahori [1J and Matsumoto [HMJ as follows: Theorem 7.2.28 (1wahori, Matsumoto):
HK(q)
has
K-basis
{a\'Ilw
H} ,
€
where aw = indS(q)w.b,,)(w)b(q) , for all w € W. al = b(q) is the identity element of HK(q) . For any 1 ~ w € W, if 'II = w.••• w. '1
is a reduced expression for w then aw = aw. ... aWe '1
generated, as
's
HK(q)
is
's
,I ,
K-algebra with identity element al '
and the following are defining relations for HK(q)
by {aw.
\'i.
€
-R}
in terms of those
generators: if
for all
w,v
€
Wand wi
€
~(wv)
= ~(w)
+
~(v)
,
R.
o
The Generic Algebra Let A = Q[u i ; wi
denote the polynomial ring over the rational
~J
€
field Q with generators {ui}w. R' such that ui -
,
= uj whenever wi
and Wj are conjugate in W . (J. Tits):
Definition 7.2.29 Coxeter system
and basis
{Tw
TwTv = Twv T2 w.
=
,
for 'a 11
v,w
H
of the
is the associative algebra over the ring A,
(W,~)
with identity T,
The generic algebra HA(u)
I \'i
€
H} satisfying: if
, + (u.-l)T , W.,
U.
,
and W·
€
R
~(wv)
= ~(w)
+
~(v)
- 108 -
A specialization is a ring homomorphism f a field.
can be regarded as
a
A+ a
,
with a
(a,A)-bimodule.
Let Hf,a = a Q HA(u). Hf is called a specialized algebra A It is a a-algebra with a-basis {l Q Tw = Tw,f I w E W}
Defi ni tion of HA(u).
and the structure constants of Hf ,a are obtained by applying f structure constants of the generic algebra HA(u) . Note:
to the
Any specialization f: A + a induces a ring epimorphism
f : HA(U)
+
Hf,a'
Examples: 7.2.30
such that f(T w) := Tw,f for all
w E W.
(i)
Let fq : A + K (K is the field of characteristic c. o in the p-modular system (K,R,F)) be the map given by fq(u i ) := q , , 'V 1 ~ i ~ t . Then Hf K = HK(q) as K-algebras (see 7.2.27). q' A + K be the map given by fl(u i ) = 1 , 'V Then Hfl ,K = KW as K-algebras. (ii) Let fl
For the remainder of the section we will fix the following
Notations: notations:
, ,
~ := Q (u.;w.
E
-R),
the quotient field of A . 'V
L = finite extension of A which is a splitting field for 11
= integral
closure of A in
HK(u) .
K.
Any specialization f: A + K can be extended to a homomorphism f*
11
+
K (see for example [CRIIJ, lemma 68.16) ..
The specializations used in [ClK] appear in
fl
and fq'
defined in 7.2.30,
have been
to parametrize the irreducible characters of G(q) which
lG(q) B(q)
The following theorem given this parametrization in
terms of the irreducible characters of W.
- 109 -
Theorem 7.2.30
(Curtis, Iwahori, Kilmoyer [CIK]):
There exists a bijection of W to
Let G(q)
from the set of irreducible K-characters
~ + ~O
the set
{~ol~o is irreducible character of G(q) ,
by 7.2.30
SW
o §7.3 Some Contravariant Forms On Y
x
Let G = (G,B,N,~,U)
be a finite group with split BN-pair of
characteristic p. We assume that G has an involutory anti-automorphism 2 8:G + G (i.e. 8(glg2) = 8(g2)8(gl) for all gl,g2 € G and 8 = id G with the following axioms: (i) '8(h)=h
forall
h€H.
(ii) e(X) = X-a for all a G associated with the root (iii) 8(N) = N, 8(n)H
= n-1 H
for all
Since U =
II X ...+ a a€..-
a
a
€
€
4>
4> ,
when Xa
is the root subgroup of
(see §1.2).
and 8 induces a map on W= NIH such that n
€
N • -1
, it follows from (ii) that 8(U) = wo UwO
=U
.
- 111 -
If G =
is a finite Chevalley group
k of characteristic p,
with its standard split BN-pair (see [RS1], p.35), In fact e,
then G, has such e .
in this case, is given by
Let V and X be two RG-lattices.
Definition 7.3.1
form 8: V x X + R is called contravariant
An R-bilinear
(or e-contravariant)
if
8(gv,x) = 8(v,e(g)x) for all
v
€
V,
X € X,
and 9
€
G.
denote by Ve the R-module HomR(V,R) , regarded
If V € mod RG,
as left RG-module by the rule (gf)(v) .- f(e(g)v)
7.3.2 for all R-module,
f
€
Ve ,g
G and v
€
HomR(V,R)
€
V
Note that since V is a free R-module
is also a free
([SL], p.343)',
hence
e ,. s an RG-l a tti ce.
V
Remark 7.3.3
The rule 7.3.2 gives a left G-action on Ve , since
e is an anti-automorphism.
If we replace e(g)
RG-module V* (see [CRII],
get the usual "dual" or "contragredient" p.245). v
€
Note also that V ~ (Ve)e
V, where a
v
€
(Ve)e
av(f) := f(v)
in 7.3.2 by 9-1 ,we
under the RG-isomorphism v + av ' is given by
for all
f
€
Ve
- 112 -
Lemma 7.3.4
Let V, X € modoRG and let a : V x X + R be any R-bilinear form. Let fa : V + XS be the corresponding R-map given by fa(v)(x) := a(v,x) , for all
v
€
only if fa Proof
and all
V,
x
Then a is contravariant if and
X
€
is an RG-homomorphism.
Suppose S is contravariant.
For a given 9
G and v
€
E
V
we have fa(gv)(x) = a(gv,x)
= a(v,s(g)x) =
fa(v)(e(g)x) by 7.3.2,
= (g fa)(v)(x) for all v
€
x
€
X.
Hence fa(gv)
V , and so fa
then, given 9
E
is an RG-map.
G,
a(gv,x)
= gfa(v), for all 9
V
€
V,
Conversely if fa
and x
EX,
E
G and all
is an RG-map
we have
= fa(gv,x) = fa(gv)(x) = gfa(v)(x)
since fa
is an RG-map
= fa(v)(S(g)x) by 7.3.2 = a(v,s(g)x) Hence a is contravariant.
o
If ·{vl' ... ,v r } and {xl' ... ,x t } are R-bases for V and X, respectively, then the contravariant form a : V x X + R of 7.3.4 is
- 113 -
completely determined by the values e(v.,x.), 1 , J Let M(e)
denote the r x t matrix whose
Definition 7.3.5
~
i
~
r , 1
(i,j)-entry is
~
j
~
t
e(v.,x.) . , J
Let V , X modoRG and let e: V x X + R be an
R-bilinear form.
Suppose rank V = rank X = t
say. Then e is said
to be non-singular if the
txt matrix M(e) is invertible in R , equivalently if the map fe: V + Xe is an isomorphism of R-modules. x In this section we will show that the set {Y, X E Hom(H,K )} X
is closed under the e-duality. FG-modules M(x,J) Let
X
~
E
are self e-dual. x
= Hom(H,K ). If S is a transversal of {gB; 9
then it is clear that {s[UJe RG-lattice Y
X
Lemma 7.3.6
We will also show that the simple
X
; s
E
S}
E
G}
forms an R-basis for the
(= RG[UJe ) .
X
Let S be a transversal of the set {gB; 9
Then {e(s) -1 (w O) ; s
E
S}
E
G}
is also a transversal of {gB; 9
E
G} .
If 9 E G then 9 = sb for a unique s E S and a unique b E B , and e (g - 1 ) = e (b -1 s -1) = e (s -1) (b - 1 ). Since e (B) = w Bw- l o 0 1 1 1 e (b - ) = (w0) b (w0) - for some b' E B, and so e(9- ) = e(s-l)(Wo)b ' (wo)-l, Proof
I
-1 -1 )(wo ) = e(s )(wO)b ' . Since the map 9 + e(g )(wo ) is a -1 bijection of the elements of G it follows that sb + e(s )(wo)b ' is
hence e(9
-1
also a bijection of the elements of G.
Hence every element 9
E
G has
the form for a unique s and a unique b'
E
€
S
B . Therefore {e(s -1 )(w ); s o
€
S}
is a transversal
- 114 -
of {gB; 9
G}
€
o
Proposition 7.3.7 :
Y x Y X woX
X 0 woX
= x(b)
x(b)
then
.
= (wo)-'S(h ' )-'s(u l fl (wo ) = (wo)-'S(h,)-l(wo)(wo)-lS(u' )-l(wo )
€
UH .
\I - 115 -
woX((wo)-la(b' )-l(wo )) = woX((wo)-la(h,)-l(wo))
= wox ((wo )-1 hi -1 (w o ))
Therefore, 0 wox
= x(b)x(b ' )-1
B
if gg,-l I. B , which shows that
is well-defined.
(ii) We show that
is contravariant.
is a-contravariant.
If
if
g,-lglg
if
g,-lglg I. B
E
B
g,g',gl
E
G then
- 116 -
(iii) {gB ; g
E
Non-singularity. G}
Let S be a transversal of the set
By 7.3.6 {e(s)-l (w )[UJa ,s o wox For S,SI
E
S
E
S}
is an R-basis
we have
t B
if sl-l
But sl-ls
for all
B s8 = siB S = Sl.
E
S,SI
E
S,
and so the
lsi
x
Therefore we have
lsi matrix
M«
,»
is invertible
hence is non-singular. Remarks 7.3.9
(i)
It follows from 7.3.7 that there exists an RGinduced from the contravariant form
isomorphism f< , > < , >, given by f< for all
XE
(i i)
o
>(x)(y):= Y and y X
E
If k E {K,F}
Y woX then the contravariant form
< , >
defined in,7.3.7, will induce a contravariant form < , >k
: Y
X
kY
X
x
x
Y
woX
kY
woX
It is clear that the form < '>k is non-singular. Let (x,J)
be an admissible G-pair and let M(x,J) be the corres-
ponding simple FG-module. M(x,J)
In ([C2] , Theorem 6.15),
is completely determined by its unique B-line
Curtis shows that (see 4.2.5) and
the parabolic subgroup GJ which is the full stabiliser of that line. \
+
R,
+
k
- 117 -
, ,
We will assume that the Coset representatives {(w.),w. chosen such that (w,.)
, -,
€
for all
e
tV
,
w.
€
R.
€
-R} are
Such choice is
possible by 1.2.4(i). M(x,J) = M(x,J)
Theo rem 7. 3. 10 G-pairs Proof
as FG-modules, for all admissible
(x,J). Let M= M(x,J)
simple FG-module.
It is clear, since
(Me)e
= M, that Me is
Let F.m be the unique B-line of M.
From ([C2J,
Theorem 4.3), we have M = FG.m = (FU-).m
= F.m m!(FU-)m ,
7".3.11 where !(FU-) follows:
is the radical of FU
If ml
€
decomposition of ml
For every u
€
M then
~(m')
. Define the F-map
~
: M+ F as
is the coefficient of m in the
according to 7.3.11,
that is
U we have
e(u)ml = ~(ml )e(u)m + e(u)x l = ~(m')«e(u)-l)m + m) + e(u)x l = ~(ml)m + ~(m')(e(u)-l)m + e(u)x l . Since e(u)
€
that (e(u)-l) Hence
U
and U-
€
!(FU ).
is a p-group, it follows
(see [CRIIJ, Thm. 5.24)
It is also clear that e(u)x l
€
!(FU )m .
- 118 -
and so
Ull
= 11
for all
u e: U.
If
h e: H then, since
hm
= x(h)m
,
we have hm'
= x(h)lJ (ml)m + hX 1
hX 1 e: x:.(FU-)m,
and
for all
h
E:
H.
that Me ~ M(x,T)
since
H normalizes
It follows that for some
F.11
U- .
is the unique
F.11.
h11
= x(h)
B-1ine of Me
and
T ~ P(x) •
To comp1 ete the proof we need to show that of
Therefore
GJ
is the full s tabi 1ize.-
Let Wi e: J and let Xl e: x:.(FU )m
be any element of M .
We have
7.3.12
Since
, -,
(w,.) e: ,
e((w,.)) e: , -,
(note that
it follows from axiom (ii) of e U.,
= Ua. ).
,
By 1.2.4(3),
that
we then have
e((w )) = (wi)h for some h e: Hi . Since GJ is the full stabilizer of i F. m and X/Hi = 1 , it follows from 7.3.12 that e((wi))m '
= l1(m')x(h)m = l1(m')m
It will then follow that
+ (w i )hx 1
+ (wi)hx l • (Wi)ll = 11
if we show that
7.3.13
Since 7.3.14
H normalises
U
and
hm
= x(h)m,
we have
- 119 -
which follows from 1.2.8, where
and
w. w., - - w0, (U) w. = U n (U ) =U n U
Since U
,
is spanned by the elements v-l , v where vl
E
+ (vl-l),
(U-);i
and v2
E
(U-)~i
E
is a p-group, !(FU-)
U- , write v = vl v2 ' . Then (v-l) = (v l v2-1) = vl (v 2-1) + U-.
If v
E
hence -1
(wi)(v-l)(w i )
= (w i )(v l (v 2-1) + (vl-l))(w i )-1 = zl(z2- 1 ) + (zl-l) ,
-1
-1
where zl = (wi)vl(w i ) and z2 = (w i )v 2(w i ) w. = (U-) , n U ~ U, and so (z2-1~m = 0, hence
. But z2
E
- -1 (w.)( (w . ) , U) w.,
,
-1
(wi)(v-l)(w i ) m = zl(z2-1)m + (zl-l)m
= (zl-l)m Therefore (since
E
!(FU )m .
(wi)m = m) ,
--1 (w,.)r(FU )m = (w.)r(FU )(w.) (w.)m -
, , , = (w., )r(FU )(w.) , -1 m -r(FU - )m and so (wi)p = P for all , ~
which proves 7.3.13
is the full stabilizer of F.p , J On the other hand, since
W. E J .
~
T. we have M(x,T) e 'V= M(x,J)
and so by reversing the above argument we get T ~ J. so M(x,J)e
Since GT
Hence T = J and
= M(x,J). This completes the proof of 7.3.10.
o
=
- 120 -
Remark 7.3.16
The functor
(,R) : modoRG
+
mod R is
hence it commutes with direct sum of RG-lattices. since k Q (X,R)
'V
= (kX,k)
R-linear,
Note also that,
(X is any RG-lattice and k
E
{K,F}) ,
R
it follows that s-dualizing commutes with the functor k Q
: modoRG
+
R
mod kG . For every J s
B.,
1et
w oJ s
is an admissible G-pair then so is w (x,J) + (wox ,oJ) is a bijection.
and if (x,J) mapping
B.
(see [HS], p.3S), wo (wox, J) and the
we have Y = Lffi Y(x,J). By 7.3.9(i), x J;:P(x) s-dual functor commutes with direct sums of
Recall that for every X E yS ~ Y and so, as the wox X
RG-lattices, we then have
Using Krull-Schmidt theorem for RG-lattices
(see [CRII], p.620), we
then have 7.3.17 for a unique S;: P(wox) . Proposition 7.3.1S
w Y( x,J) ~ Y(wox, oJ)s as
admissible G-pairs
(x,J).
Proof
RG-lattices, for all
From 7.3.17, we have
7.3.19 for some S;: P(wox).
By taking the s-dual of 7.3.19 we get
- 121 -
e 'V
and, by remark 7.3.16, we have FY(x,J) = FY(wox,S). To show that w S = oJ, it is enough to show that Soc(FY(wox,S)) ~ M(x,J) (see [HSJ, theorem 3.10).
But since the head of FY(x,J)
is M(x,J) and
by 7.3.10, we have
Hence S
=
w oJ,
and so
o Now let X E ~ and let {X., i 1
E
I}
be a set of RG-1attices
such that {KX i , i E I} is the full set of simple KG-components of KY . Since KY ~ KY (see 5.0~4(ii)), we can identify the KGX X wox components of KY
X
RG-isomorphism f
wox
. On the other hand the
defined in 7.3.9(i),
induces a
KG-isomorphism >
Therefore"for each i that
(KXi)e
E
I
, (KX.1 )e 'V= KX.1
is simple since
commutes wi th di rect sums).
I
for a unique i
7.3.20
E
I (note
((KX.)e)e ~ KX. , and the e-dua1 functor 1 1 Since
,.
((KX.)e)e 'V= KX. , 1
the map i
defines an involution on the index set I. Let (x,J)
I
be an admissible G-pair,
and suppose that
+
i
I
- 122 -
and 7.3.21
where d. ( J) and d w 1, x, i,(woX' OJ)
€
~~O.
By applying the 6-dual functor to 7.3.21 we get
7.3.22
w (KY(woX' OJ))6
'" =
L
id L
id
(KX. )6 d w 1 i,(woX' OJ) d w KX" i,(woX' OJ) 1
By 7.3.18 and remark 7.3.16 we have 7.3.23
as
KG-modules.
Therefore, for each
d
W
i , (woX, oJ)
I
€
= [KX"
by 7.3.22
1
= [KX.1 7.3.24
i
I
by 7.3.23
KY(X,J)J
= d"1 , ( X, J)
Using dual argument to the above, we also have 7.3.25
Now consider the decomposition matrices sys terns Si nce
(KE
X
,E
X
,FE) and X
(KE
woX
,E
woX
,
of the and D woX respectively.
the set I indexes the rows of D and D X woX
- 123 -
On the other hand
Therefore the set of simple FE -modules
x
and the set of simple FE
wox
-modules are in 1-1
Consequently D and D x woX
correspondence.
have the same size, namely
III
x
2 IP (x)1 •
Moreover, using 7.2.3,
it follows from 7.3.24 and 7.3.25 that D and x have the following forms
.
J
i;" odi .(x.J)
D X =
with the D
x
)\
i' \oodi,.(x. J ) i-th row of D woX
•
(i
E
I)
identical to the i'-th row of
Summarizing the above we have
Theorem 7.3.26 components of KY
Let X E ~, X
and suppose that the set of simple KG-
are indexed by the finite set I.
be the involution on I defined above. J
~
Let i
Then for every i
E
-+
i'
(i
I and every
P(x) ,
=d
wand d. , ( J)
ii, (w ox, oJ)
1 , X,
Consequently, the index set I can be arranged so that D and D woX X are' ide nti ca 1.
E
o
I)
- 124 -
§7.4 The Direct Product of Split BN-pairs In this section we consider the direct product of two finite groups with split BN-pairs. We will show that such product also has a split BN-pair, and will derive some results which are relevant to the subject of chapter 7.
In particular, we will prove some results concerning the
decomposition numbers
d.1, (X, J)
applied later in §7.5
to the case of the general linear group.
(see 7.2.3).
group with a split BN-pair of rank ii
These results will be
and characteristic p,
prime p
>
o.
G = G(l)
x
G(2) be the direct product of the groups
Write H(i) = B(i) n N(i)
and Wei) = N(i)/H(i)
G(i),
E
i
=
1,2.
Suppose G = G(l)
R(i),U(i)),
i = 1,2,
N = N(l) (W = W(l)
N(2)
x x
x
is a finite group with a split BN-pair of
for some prime p
W(2) , B. = B.(l) ~ B.(2))
>
o.
Then B = B(l)
= l
, 1) ,
So we may identify
W(l)
with
j
{(W3l),1)
;
W(2)
(3)
Suppose
r
€
R = ~(l)
U~(2),
r = w(l)
€
J
R(l),
-
;
(1) (2) of W x W
with
i
= 1,2
j
The subgroup (1) W
.
= 1, ... ,R. l > , and ~(l)
Similar identification can be done
•
and in this way we have ~
~ and w = (w(l) ,w(2)} we may assume that then identifying
.
is isomorphic to
«w~l),l);
= 1, •. . ,R.,}
and ~(2),
for
Therefore
~(i) = {W~i), ••. , wi~)}
Suppose that «w j
N(l) x N(2) ,
9
r
= ~(l) U ~(2) .
€
W(l) x W(2).
€
~(i),
w(l) J
with
i
Since
= 1,2.
(w~1),,) J
€
Suppose W(l) x W(2)
we have rBw
= (w (1 ) , 1) ( B( 1) ,B ( 2 )) (w (1 ) , w( 2 ) ) J
=
(w31) B(l )w(1), B(2)w(2))
~ (B(l )wjl )w(' )B(l) ~ B(l)w(l)B(l) ,B(2)w(2)) ,
by the
BN-pair axioms of
= (B(l)w(l)w(1)B(l) ,B(2)w(2))
G(l) .
~ (B(lJw(l )B(1) ,B(2)w(2))
J
~ (B(1)w~1)w(1)B('),B(2)w(2)B(2)) ~ (B(1)w(l)B(l),B(2)w~2)B(2))
= (B(l) ,B(2) )(w(l )w(l) ,w(2))(B(1) ,B(2)) J
= BrwB u BwB .
- 126 -
The case when r E R(2) (4)
For rEB.,
is similar.
using similar discussion to that in (3) and the
BN-pair axioms of the groups that (5)
rBr
G(i);
i = 1,2, one can easily see
f B.
Finally we have B = B(l) x B(2) = U(l )H(l) x U(2)H(2)
= (U(l) x u(2)) (H(l) x H(2)) . Therefore, putting U = U(l) x U(2)
and H = H(l) x H(2) , we then
have U ~ Band B = UH . It follows from (1)-(5) that (G(l)xG(2), B(l )xB(2), N(l )xN(2), B.(l) U R(2), U(l )xU(2)) form a split BN-pair in G = G(l) x G(2) of rank tl + t 2 . It is also clear that this split BN-pair is of the same characteristic as (G(i),B(i),N(i),B.(i),u(i)),
i = 1,2.
o
the lemma. Let (K,R,F)
be a p-modular system such that K (and hence F)
is a splitting field for G(i),
Xi
€
This completes the proof of
i = 1,2, and all its subgroups.
x If x, E f( = Hom(H,K ), then x has the form (Xl,X2)' where x f((i) = Hom(H(i),K ), i = 1,2. The value of X (= (Xl'X2)) or
h = (h l ,h 2)
€
H is given by
Any X = (Xl,X2)
€
~ can be extended to a character xB of B by putting
- 127 -
where
i = 1,2,
\B(i)'
is the extension of Xi
to a character of
B(i) which has the trivia' value on UCi ) . and for e = Xi
where
IH(i)I- l
h EH ( i)
x
for all
(9,9) 1 2
Let V = RG[UJe X
=
X
1,2 , let V(i) xi
=
RG(i) [u(i) Je
G(2))-module
~
IHI- l
E
X
G(1)
~
g2 x2 '
G(2 ) and all
x
, where G = G(l)
x
x 1
vCl) , x xl 2
E
G(2),
U = U(l)
E
v(2 ) . x2
x
U(2)
and
x(h-')h .
hEH
y ~ Vel) ~ V(2) as RG-lattices X X, R x2
Lemma 7.4.3
Proof
E
x2) := glx l
Let n: RG
n( (91,92)) := 91
~
RG(l) ~ RG(2)
+
92'
for all
be the
(91,92)
E
R-map given by G.
n is clearly an
R-isomorphism.· It is also an RG-isomorphism where the G-action on RG(l) ~ RG~2)
is the one 9iven by
R
for all
(91,92)
that if 9
E
G,
= (91,9 2)
E
xl
E
G,
9 [u(l)Je ~ 9 [u(2)Je ., xl 2 x2
E
xi
has
(see [eRIIJ, §lOE), where the
G(2)-action is given by: (gl ,g2)(x l
e
1
x
=
The R-lattice Vel) ~ V(2) Xl R x2
x.(h-l)h.
E
a structure of left R(G(l) G(l)
i
RG(l),
and x2 E RG(2). It is clear then n sends 9[UJe E RG to x RG(l) ~ RG(2) . Therefore
- 128 -
n(y ) = n(RG[U]SX) X ~ RG(l)[U(l~ S
9 RG(2)[U(2)]S xl R x2
= y(1) Xl
9 y(2) R x2
Hence Y ~ y(l) 9 y(2) x2 X Xl
as
RG-1attices.
o
It follows from 6.0.7 that we have a decomposition
m Y(x,J),
'V
y = L
7.4.4
JsP(x)
X
of Y
X
into mutually non-isomorphic indecomposable RG-lattices.
w~i), i
=
1,2,
be the unique element of W(i)
Then it is clear that w = (w(l) w(2)) o 0' 0 element of W(l) Let w' J
1
~
E
W(l)
is the unique
J
= (w(l )wP) w(2)) . Hence oJ'
0
= (U~l), 1) J
Therefore -J
W(2)
B. (= B.(l) ~ B.(2)) , and suppose that w. = w~l) J
U.
x
of maximal length.
W(2) of maximal length.
x
Then, identifying w~l) with
j ~ R.1 •
and so
E
Let
w.
= U.J J
(w~l),l) J
E
J
W(l)
x
E
R(l) ,
-
W(2) , we
- 129 -
7.4.5
=
(H\l), 1) . J
Similarly, if Wj = W3 2) € .8.(2), 1 ~ j ~ R,2'
then
7.4.6 i = 1,2.
Lemma 7.4.7
(ii)
.
is an admissible G-pair if and only if J = J l U J 2 ' J c R(2) and (x.,J.) is an admissible G(i)-pair,
(x,J)
say J 1 -cR(l) ' i'=1,2. Proof
The n
2 - -'
1
1
Both (i) and (ii) follows from 7.4.5 and 7.4.6.
o
Let X = (xl,x2) € ~.
Since Y ~ Y ~ Y as RG-lattices, it X xl R x2 follows that E(Y ) = E(Y ) ~ E(Y ) (see [eRII], Lemma 10.37). To X Xl x2 define an explicit algebra isomorphism between E(Y) E(y(l) ~ E(y(2)), Xl R x2 E(y(i))
Lffi.
=
n.€N(1) ,1
where An (n EN)
and
X
suppose that E(Y)
R.A(i), ~
i
=
1,2
=
Lffi
n€N
x ex
N(2)),
and and y(i) = RG(i)[u(i)J)
are defined as in §4.0.
and consider the,diagram ~
)
vel)
)
A(l) ~ A(2) nl n2 vel) ~ y(2)
7.4.8
1
Y
n
(here Y = RG[U]
and A(i) n.1 (n.1 € N(i))
n = (n ,n 2) E N (= N(l) l
R.A
y(2)
Let
- 130 -
where a : V + V(l) Q V(2) to [U(l)] Q [U(2)]
is the RG-isomorphism which sends
(in fact a is induced from n: RG
defined in the proof of 7.4.3).
+
[U]
RG(l) Q RG(2) ,
We have
aAn([U]) = a([UnU])
= a ([ (U ( 1) , U( 2) ) (n ' n ) (U (1) , U(2 ) ) ] ) l
2
= a([(u(l)nlu(l) , u(2)n u(2))] 2
=
a(([u(l)nlu(l)] , [u(2)n U(2)]) 2
= [u(l)nlU(l)] = (A(l)
Q
n
Q [u(2)n u(2)]
2
A(2)) a ([U]) . n2
l
Therefore aA = (A(l) Q A(2))a, n
Lemma 7.4.9
n
The map A(
and so the diagram 7.4.8 is commutative.
n
l
2
nl ' n2
)
+
A(l) ~ A(2) nl R n2
defines an R-algebra isomorphism between E(V)
(n. '
E
N(i),
i = 1,2 )
and E(V(l)) Q E(V(2)) R
Proof
The R-algebras
since V ~ Vel) Q V(2)
E(V)
and E(v(l)) Q E(V(2))
are isomorphic,
The lemma follows from the commutativity of
R
o
diagram 7.1.8. We now return to the R-orders
E(V ) x
~
= E =e x
x
E(V)e
X
=
E(V) x
L
~
R.e A(w) , X
WEW X
and, for
i = 1,2 ,
and E(v(l)} Q E(V(2)) xl R X2
,
- 131 -
and W ,w(i)(i=1,2) x xi
where
are the subgroups of W, Wei) , which stabilize x, xi ' respectively. If w = (v l ,v 2)
E
W then wx = (v X1,v X2) l 2
Proposition 7.4.10
The map exA(w)
and so W = W(l) x Xl
x
W(l) X2
e A(l) ~ e A(2) (w = (v ,v 2) 1 Xl (v l ) X2 (v 2)
+
defines an algebra isomorphism between E and E ~ E x Xl R x2 Proof
E = ~@ R.exA(w)'
We know that the R-orders
X
WEW
and
X
are isomorphic as
The result follows since, for every w = (v ,v ) 1 2
E
W = W(l) X xl
R-algebras.
x
W(2) x2
the commutative diagram of 7.4.8 will induce a commutative diagram a
V
>
X
7.4.11
exA(w)
1
(11
a
)
X
Y
Xl
Now suppose that (x,J) E
Xl R X2
1
Y
X = (Xl,X2) and X.1
VIiI 9 V(2)
H( i)
e A(1 ) 9 e A(2) x2 (v 2) Xl (v l ) (2)
~
Y
is an admissible ,
0
X2 (G(l)
x
G(2))-pair, with
i = 1,2 . By 7.4.7(ii), we may assume
E
W) x
- 132 -
that J = J l U J 2 , where J i ~ P(Xi)' i = 1,2. Let M(x,J) , M(x.,J.) be the simple F(G(l) x G(2)) -, FG(i)-module, which
, ,
corresponds to the pairs (see
§4.2).
(x,J) and
(xi,J i ),
i = 1,2,
respectively
It is well-known (see [eRII], Thm. 10.33) that
M(xl ,J l ) 9 M(x2,J 2) is simple F(G(l) F F is a splitting field for G(i),
x
G(2))-module. Moreover, since
i = 1,2 , every simple F(G(l)
x
G(2))_
module is of the form M(xl,J l ) 9 M(x2,J 2) for some admissible G(i)-pair F
M((xl,X2),J l G J 2) ~ M(Xl ,J l ) 9 M(X2,J 2 ) F
Proposition 7.4.12 F(G(l)
x
G(2))-modules, for all admissible
as
(G(l)
x
G(2))-pair
that
I (.)(M(x.,J.))
((Xl'x2)' Jl U J 2) . Proof
We have seen in (§4.2, Thm. 4.2.5) .
one-dimensional of E(Fy(i)),
U'
right E(Fy(i1-module affording the character w(xi,J i ) i = 1,2. That is I (.)(M(x.,J.)) = F.m. , where U '
m.
,
€
M(x.,J.),
0
"
,
and
"
mi
is a
"
( i) Ah,F
=
for all
h
€
H(i) ,
and ,
7.4.13
mi
A(i)(i)
0
(w.
),F
J
for all
w(i) J
(i) if w. '- P(x·)\J· J "
0
€
R(i)
-
={
,
-m.
(')
if w.' J
€
P(x·)\J. "
.
Since M(xl,J l ) 9 M(X2,J 2 ) is a simple F(G(l) F
x
G(2))-module, it
- 133 -
follows from theorem 4.2.5 that I U(M(X1,J 1 ) Q M(X2,J 2 )) F
= I (1)(M(X1,J 1 )) U
is a one-dimensional right E(FY)-modu1e.
~ I (2)(M(X2,J 2))
U
To complete the proof it is
.
enough to show that F.m 1 ~ m2 affords the character ~((X1'X2),J1 u J 2 ) of E(Y). Let n = (n ,n 2) E N ~ N(l) x N(2) . Recall (from 2.2.13) 1 that the action of A~~~F E E(Fy(i)) on mi € IU(i)(M(Xi,J i )) is given by
m.1
0
(i) _ An., F - an. m.1 1
,
1
where Si nce 7.4.14 it follows that (see 7.4.8)
=
(a
n
1
~
a
n2
)(m
1
7.4.15 It is clear that, for all
h = (h, ,h 2 )
E
H,
~
m2 )
- 134 -
,
Let w.
€
-R = -R(l) ~ -R(2).
,
,
wP) with
(wP), 1) ml 9 m2
0
€
,= w~l) , -R(l)
If w.
W(l)
x
A(w.),F ,
W(2) , we have
= ml 9 m2 0 A (1) ((w. ),l},F
,
= ml
0
= ml
0
A (1) 9 m2 (wi ),F (wi
by 7.4.15
Al F '
),F
if w~l) i P(Xl)\J l if w~l) e P(Xl)\J l
, = w~2) ,
Similarly, if w.
ml 9 m2
0
€
-R(2) , 1
={:,
,
A(w.),F
Therefore, from 7.4.16 and 7.4.17,
~((Xl
0
9 m2
A (1)
7.4.16
7.4.17
, then, identifying
€
,X2),J l
. u
J 2)·
~
i
~ ~2
'
then we will have
if wi(2) I. P(X2)\J 2 9 m2
if w.(2)
,
€
P(X2) \J 2
it follows that
Hence, by theorem 4.2.5, we conclude that
M(Xl ,J l ) 9 M(X2,J 2 ) ~ M((Xl'X2),J l ~ J2) F as
F(G(l)
x G(2))
-modules.
Now let x = (Xl,X2)
o tV
H and let
7.4.18
ID Y = L; Y(x,J), x JcJ(x)
7.4.19
y(i)
xi
=
L;ID V(x.,J) J~P(Xi) ,
,
i
= 1,2,
-----; I
- 135 -
be the decomposition of Y
X
(1)
and Y
into a direct sum of mutually
Xi
non-isomorphic indecomposable RG- , RG(i)-lattices, respectively, according to 6.0.10. Proposition 7.4.20
Let X = (xl,x2)
an admissible
x
(G(l)
To prove 7.4.20,
G(2))-pair.
of characteristic p.
'H"
and suppose that
If S = Sl
U
S2'
(x,S)
is
then
we need the following
Let G = (G,B,N,B.,U)
Lemma 7.4.21
€
Let (x,S)
be any finite group witha split BN-pair be an admissible G-pair, and Vex,S)
be the indecomposable RG-summand of Y = RG[U] which corresponds to (x,S).
Then .",
E(FY(x,S))/!(E(FY(x,S)) = F Proof
Let e
= e(x,S)
be the primitive idempotent of E(FY) which
corresponds to the indecomposable FG-summand FY(x,S) of FY.
Then we
have E(FY(x,S)) '"= eE(FY)e
(see appendix, lemma 1), and hence
!(E(FY(x,S)) ~ e!(E(FY))e
(see [CRII], Prop. 5.13). The right E(FY)-
module eE(FY)
is a projective cover of the one-dimensional right E(FY)-
module S(x',S) which affords the character 1jJ(x,S) eE(FY) I e!(E(FY))
(= '" Sex,S))
(see 4.2.4).
Hence
is also one-dimensional.
. The map
X
€
eE(FY),
is clearly an F-isomorphism between eE(FY) I e!(E(FY))
- 136 -
and eE(FY)e I e!(E(FY))e.
Therefore 'V
E(FY(x,S))/!(E(FY(x,S))
(=
eE(FY)e I
e~(E(FY))e)
is one-dimensional, hence E(FY(x,S)) I
~(E(FY(x,S))
'V
=F .
o is an
It is clear that Y(x1'S,) ~ Y(X2,S2) . R R(G(l) x G(2))-summand of y(l) ~ y(2) (~Y). Proof of 7.4.20
(i) We first show that Y(x"S,) ~ Y(x2,S2)
'V
F ~ (Y(x1'S,) ~ Y(X2,S2)) F R R is indecomposable. Consider the endomorphism algebra
lattice by showing that FY(x, ,S,)
E(FY(x1'S,)
~
F
~
is indecomposable R(G(l)x G(2))_
FY(x2,S2)
'V
FY(x2,S2)) = E(FY(x"S,)
(see [CRII], '0.37).
~
F
(=
E(FY(x2,S2))
The idea'
M= ~(E(FY(x"S,))
~ F
E(FY(x2,S2)) + E(FY(x, ,S,)
~ ~(E(FY(x2,S2)) F
is a nilpotent idea' of E(FY(x, ,S,))
~
E(FY(x2,S2)'
hence
F M~
r(E(FY(x"S,))
-
E(FY(x"S,)
~
~
F
E(FY(x2,S2))'
On the other hand
E(FY(x2,S2) I M
~ E(FY (xl'S, ))/!E{FY (xl'S,) ~ E(FY (X2 ,S2) )/~(E (FY (X2 ,S2))
(see [CRII],
proof of '0.38)
'V
= F ~ F 'V= F , by 7.4.2'
is semisimp'e. Therefore, M contains r(E(FY(x, ,S,) -
~
F
- 137 -
hence M= !.(E(FY(Xl ,Sl)) E(FY(Xl,Sl))
6a
E(FY(X2,S2)).
6a
E(FY(X2,S2)) / M~ F,
Since
it follows that
F
(~E(FY(Xl ,Sl) 6a FY(X2,S2))
E(FY(Xl ,Sl) 6a FY(X2,S2))
is local algebra,
F
and so FY(Xl,Sl) g FY(X2,S2)
is indecomposable F(G(l)
x
G(2))-module
F
(see [CRIIJ, Cor. 6.4). R(G(l)
x
Consequently Y(Xl ,Sl)
6a
Y(X2,S2)
is indecomposable
G(2))-lattice. Q)
'V
(ii) We have Y = L Y(x,J) X J~P(x) ~
Vel)
g
xl
Since Y(Xl,Sl) 6a Y(X2,S2)
y(2) x2
by 7.4.3
is an ~ndecomposable R(G(l)
it follows, using Krull-Schmidt theorem, that Y(Xl ,Sl) for a unique J
~
-P(x).
x
G(2))-lattice, Y(X2,S2) ~ Y(x,J)
6a
But since the head of FY(Xl ,Sl) 6a FY(X2,S2)
is
F
isomorphic to M(Xl ,Sl)
6a
M(x2,S2)
F
(=
M(x, Sl ~ S2)'
by 7.4.12), we
must have J = Sl US2' and so
This completes the proof of 7. 4.20. Let X = (Xl'X2)
€
Pi'. For i = 1,2, let {xji),
o j
€
Ii} be the
full set of simple KG(i )-components of Ky(i) . Then {XP) 6a X(2) Xi' J (i,j)
12} is a full set of simple K(G(l) x G(2))-components of Ky(l) 6a Ky(2) . Since KY ~ Ky(l) 6a Ky(2), the simple K(G(l) x G(2))_ xl X2 X xl K x2 €
11
I
x
- 138 -
components of KY are also indexed by the set 11 x 12 , Let x {X ,.,.J , (i,j) € 11 x 1 } be the full set of simple K(G(l) x G(2))_ 2 components of KY x and assume that X,.,.J ((i,j) € 11 x 12) is taken to X~l) 9 X~2) , K J For (i,j)
under the isomorphism KY ~ Ky(l) & Ky(2) xl K x2 €
11
12 and J
x
~
P(x),
let d(i,j),J denote the
((i,j),J)-entry of the decomposition matrix of the system (KE , E , FE ) x x x 5i mil a r 1y, for j € 1i (i = 1, 2 ) and 5 ~ P(xi)' 1e t di , 5 de no te the
(i,5)-entry of the decomposition matrix of the system
Let x = (xl,x2)
Proposition 7.4.22
'V
€
Let 5 = 51 U 52 be a subset of P(x). de'1,J, .) 5 Proof
=
,
11
xi
,E
xi
,FE
xi
Then, for all
(i ,j)
€
11
x
12 '
we have
*
= de'1,J.) , 5 = [X 1.,J. I KY(x,5)J
,
1,2 , let ni be the K-character of G(i) afforded by 2 KY(X·,5.) . Let 8i l ) ,83 ) be the irreducible K-character of
For i 1
= 1
G(l)., G(2) , which is afforded by X~l) , X~2) , 1
the
K(G(l)
x
G(2))-module
nln2 which is given by
J
).
and 12 be as above.
d.1'1 5 . d.1'2 5
From theorem 7.2.3, de'1,J.) , 5
H,
(KE
respectively.
Then
KY(x1,5 ) & KY(X2,5 ) affords the K-character 2 1
- 139 -
X~l) ~ X~2) affords the character 8~1)8~2)
module
J
1
1
(see [eRII], Prop. 9.23(ii))
(1) (2)
But G(l) x G(2)
1 L (1) (2) IG(1)xG(2)1 (gl,g2)EG xG
IG(1)I- l IG(2)I- l
e~l)e32)«gl,g2))·T)1112«gil,g21))
L
8~l)(gl)8f)(g2)T)1(g11)112(g21)
(gl ,g2)EG(1 )xG(2)
Therefore
= [XP) 1
d.* S J, 2
* S = d.1, 1 = d. S
• d.1 S
1'1.
'2
This completes the proof of 7.4.22.
by theorem 7.2.3.
o
- 140 -
§7.5 The Case Of The General Linear Group Let G = GL(n,q), Wq
(= GF(q)),
the general linear group over a finite field
where q is a power of a prime p > O.
Let
(G,B,N,~,U)
be the split BN-pair in G defined in 1.2.5. In this section we apply the results of Chapters 6 and 7 to the case G = GL(n,q).
In particular we will see that the problem of calculating the
decomposition numbers d. ( "J) (see 7.2.3) can be reduced to the case ',"Jhen x = 1
" x,
Let
(K,R,F)
be a p-modular system such that K is a splitting
Let r '. (Fx) q q-l Let s be a multiplicative generator of
field for G and all its subgroups. in 4.1.14. i
E
{l,2, ... ,n}
(Kx) q-l
-+
F;,
be as
and for
, let '1
s.1
=
It is clear that
'1
s
i 1
'1 tV
H = B n N = x order q-l ,
for all
~
theory of abelian groups
n
n.
, where
,
is a cyclic group of
It follows from the representation
(see [CRI], p.37)
x that the set R = Hom(H,F )
of multiplicative characters of H is indexed by the set of all (al, ... ,a), where 0 n ,
~
a., < q-l , for all
i
E
{1,2, ... ,n}
n-tuples In
fact
R = h(al, ... ,a n ) where x(a 1, ... ,a n) 7.5. 1
E
I
0 ~ a., < q-l ,
for all
R is given by:
x(al, .. ·,a n)
.
( Xl
o
'2... 0 ) xn
n 1+
IT
i =1
a. x.' 1
i
E
{l, ...
,n}}
- 141 xl
for all
Xl ..
E
'x
/
H
It is sometimes convenient to identify
n
al, ... ,a n with their residue classes mod(q-l). For everyn-tuple (a1, •.. ,a n ),0::; a.1 IV
be the element of H = Hom(H,K
x
x al ,···, a)=A (A
4>A(S) = l~ '.
E
p)
. Therefore then
It follows that for every J ~ ~ ,
S
7.5.14 Writing 4>A(S)
(S ~~)
as a combination of irreducible characters
can be achieved using Young's rule.
For that we need to use tableaux with
repeated entries as the following definition suggests.
- 149 -
(See [GJJ, p.44):
Definition 7.5.15 (i)
A tableau
T is of type ].1,
,
the number i For example
occurs ].1. 2 2 1 1
for some ].1 E P,
if for each
i ,
times in T. (4,2,1)-tableau of type
is
(3,2,1 2 ) .
3 1 4
A tableau T is semistandard if the numbers are non-decreasing
(ii)
along the rows of T and strictly increasing down columns of T The set of partitions of n is partially ordered by the following If A = (1.. 1 ,1.. 2' ... ) and ].1
order:
then we write ].1
~
A provided that for all
].1.
i=l If ].1
~
) are partitions of n,
j
j
j
L
= (].11 ,].12' ...
'
LA.
~
i =1
A a nd A f].1
'
we wri te ].1
