London Mathematical Society Lecture Note Series. 94
Representations of General Linear Groups G.D. JAMES Fellow of Sidne...
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London Mathematical Society Lecture Note Series. 94
Representations of General Linear Groups G.D. JAMES Fellow of Sidney Sussex College, Cambridge
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CAMBRIDGE UNIVERSITY PRESS Cambridge London New York Melbourne
Sydn~y
New Rochelle
Published by the Press Syndicate of the Unive~sity of camb~idqe The Pitt Buildinq# Trumpinqton St~eet, camb~idge CB2 1RP 32 East 57th St~eet, New Yo~k, NY 10022, USA 296 Beaconsfield Pa~ade, Middle Pa~k, Melbou~ne 3206, Aust~alia ©Camb~idge
University Press 1984
First published 1984 Printed in Great Lib~ary
B~itain
at the University Press, cambridqe
of Conqress catalogue card number:
83-25171
British Library C&taloguinq in Publication Data James, G.D. Representations of qeneral linear qroups (London ~athematical Society lecture note series, ISBN 0076-0552: 94) 1. Linear algebraic qroups I. Title II. series ISBN
0 521 26981 4
Contents
Abstract List of Symbols
vii x
1
Introduction
1
2
Examples
7
3
Gaussian polynomials
18
4
compositions of n
20
5
Root subgroups of G
23
6
Subgroups of G
28
7
Coset representatives
30
8
Subgroups of G used for induction
35
9
Some idempotent elements of KG
10
The permutation module M A
47
11
The Submodule Theorem
56
12
A lower bound for the dimension of S
13
The Kernel Intersection Theorem for S(
14
Reordering the parts of A
80
15
The Kernel Intersection Theorem
84
16
Consequences of the Kernel Intersection Theorem
100
17
Removing the first column from [A]
109
18
Isotropic spaces
114
19
The prime divisors of Gaussian polynomials
124
20
The composition factors of S(
136
n
n
associated with compositions
n
Acknowledgements References
-
40
n
n-m,m
)
67
u
n-m,m
)
75
145
146
ABSTRACT
This essay concerns the unipotent representations of the finite general linear groups GL (q). n
An irreducible unipotent representation is,
by definition, a composition factor of the permutation representation of GL (q) on a Borel subgroup, and the ordinary irreducible unipotent n
ropresentations may be indexed by partitions A of n, as may the ordinary Lrruducible representations of the symmetric group G n• r.ature is that the representation theory of dppuurs to be the case "q
~
n
The remarkable
over an arbitrary field
= 1" ,of the subject we study here.
The most important results are undoubtedly the SUbmodule Theorem (Chapter 11) and the Kernel Intersection Theorem (Chapter 15), but there seems l.u h
5.1
GL (q)
The group of automcrphisms of V
2.4
H
The group of diagonal matrices
5
H*
A certain subgroup contained in G*
8.1
h
The number of non-zero parts of A
4
A certain diagonal matrix
5
n
r
r
The hook length of the (i, j) node in [A] I
r
The identity r x r matrix
K
A field of characteristic coprime to q
K
K extended by a primitive p where q is a power of p
1. (m) p
20.1
th
root of unity,
The least non-negative integer i such that m < pi
8
19.1
M).,
The permutation module on P
[m]
1 + q + q
lm l
[1]
n
The dimension of V
2.4
A Gaussian polynomial
2.14
2
[2] •••
+ ••• + q
A
m-l
10.1 2.5
10.17
[m]
A binomial coefficient A
parabolic subgroup
A
prime number
6
The field of rational numbers A power of a prime number It
II.
r
A subset of {l, 2, ••• , h}
10.8
The set of subsets of {l, 2, ••• , h} of cardinality r
10.8 10.20
N* 't:
A
certain subset of
., }..
.'
A
certain submodule of M A
e;,
The symmetric group on n symbols
n
'1'
A
±
u
± 1I).
V
\
11.11
The initial A-tableau
4.2
The group of upper/lower unitriangular matrices
5.6
± A certain subgroup of U
6.1
The n-dimensional vector space over If of which GL (q) is the group of automorphisms q
2.4
The group of permutation matrices
5
A root subgroup
5
n
w
An element of X, ,
5
~J
The ring of integers
Q 1(,
s. y,
Elements of f'q
15
closed subset of
5.1
I'
A
r
The "commutator" subset of r
I' (r)
{(i, j)
II
A KG
n
In;;; i
> j
-homomorphism
i;Il
S;
r
S;
n}
5 9.4
An
K
A,
ll,
\)
element of K
COmpositions of n
4
v (m)
The largest integer i such that p
11",
Permutations
p
0, T
i
divides m
19.1
11">..
A certain permutation, depending on >..
11.3
: "'~11" .
A certain permutation, depending on R
12.1
R
s
r;Il
{(i, j)
1 ~ it' j
.+
{(i, j)
1
~ i
•
are vectors in V, we let
I
v
denote the subspace of
2.5
DEFINITION. [m]
Ln
=1
l'
v
2'
•.• , v k•
If m is a non-negative integer, let
+ q + q
particular, [OJ
spanned by v
2
= 0,
+ ••• + q
1, and [m]
[1]
Note that if we put q
m-l
=
(qm - l)/(q - 1).
1 in this definition, we get [m]
~
= m. Bear
Lhis in mind while we go through a "q-analogue" of our statements about C5. • n
Consider a vector space over K whose basis elements are the 2-dimensional aubapacee of V.
Thus, for example,
hclongs to our vector space.
Since G permutes the 2-dimensional subspaces n
uE V, our vector space may be viewed as a KG -module. n
The dimension of this
lIj,Jace is simply the number of 2-dimensional subspaces of V, which, by a lIimple calculation (see Theorem 3.1) is In] [n -
1]
[2] Consider the subspace S the
S(n-2,2) consisting of those vectors satisfying
following two conditions:
( ~~ .6)
(0)
+ K
2
=a (1)
I 'lIt'
~
The sum
of the coeffic ients is zero.
(Thus, we reqUire that
in our example above, for this condition to hold.) For each l-dimensional subspace U of V, the sum of the
fficients of the 2-dimensional subspaces containing U is zero. The subspace S is a KG -module, since conditions (0) and (1) are pren
"" rved under the action of Gn • We recommend at this stage that the reader attempts for himself to , "rlstruct a non-zoro clement of S.
It is conceivable (and after some effort,
the reader might believe likely1) that S is the zero subspace; this would not be very interesting. To exhibit a non-zero element of S, we resort to the notation of
projective geometry.
A line will denote a 2-dimensional subspace U of V,
and the points on the line will be the l-dimensional subspaces of U.
EXAMPLE.
If q
= 2,
<e
<e >, and <e + e >. 2 2 l <e >, 2
<e
l
l,
e > contains three l-dimensional subspaces <e >, 2 l
In the picture below, the line through the points <e
1
+ e > is labelled +1; this denotes the fact that in our linear 2
combination of 2-dimensional subspaces, <e
l,
e > occurs with coefficient +1 2
<eo +1 -1
(2.7)
-1
-)
<e,-fe" fe3+e..>
<e,-te..>
+1
<e..;> The picture corresponds to
an element of the vector space over 2-dimensional subspaces of V.
K
whose basis eleme~ts are the
A glance at the picture shows that condition
2.6 hold, so we have constructed an element of S.
EXAMPLE.
If q
= 3,
<e
l,
e > contains four l-dimensional subspaces, <e >, 2 l
<e >, <e + e >, <e - e 2 l 2 1 2>·
-" 10 -
+1 -1
-1
-]
-1
+1
8)
+1 +1
leave
the reader to label the four points in the middle of the picture;
example, the top left-hand point is <e fmensional intersection of <e
l
+ e
4,
e
l
2
+ e
2
+ e
3
+ e
4
>, being
the
+ e > and <e + e e + e >. 3 l 4 2, 3
in, we have an element of S. Compare (2.8; q = 3), and (2.7, q = 2), with (2.2, q
= I?).
Notice that both the above Examples used four linearly independent tors e
lJ)
l,
e
2,
e
3,
e
4•
In fact, all the following statements are true for
Assume that the characteristic of K does not divide q.
Then
(i)
S is non-zero if and only if dim V