Hall polynomials for symplectic groups

87 Pages·1992·2.169 MB·English

by  Eva Zabric

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Hall polynomials for symplectic groups
Zabric, Eva, Ph.D.
University of Illinois at Chicago, 1992
U M I
300 N. ZeebRd.
Ann Arbor, MI 48106

HALL POLYNOMIALS FOR SYMPLECTIC GROUPS
BY
EVA ZABRIC
B.A. University of Ljubljana, Slovenia,1983
M.S. University of Illinois at Chicago, Chicago, 1986
THESIS
Submitted in partial fulfillment of the requirements
for the degree of Doctor of Philosophy in Mathematics
in the Graduate College of the
University of Illinois at Chicago, 1992
Chicago, Illinois
THE UNIVERSITY OF ILLINOIS AT CHICAGO
Graduate College
CERTIFICATE OF APPROVAL
October 8. 1992
I hereby recommend that the thesis prepared under my supervision by
Eva Zabric
entitled Hall Polynomials for Symplectic Groups
be accepted in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
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Advjper (ChairpersBji of Defense Committee)
I concur with this recommendation rU^—- La 7 <5"~^ <y
Department Head/Chair
Recommendation concurred in:
r<f&T~-Oy" .
Members of
fij uJ/jf' Thesis or
S R ~ Dissertation
Defense
Committee
• The University of Illinois
| at Chicago
ACKNOWLEDGMENT
I would like to thank my advisor, Prof. Bhama Srinivasan, who suggested this dissertation
topic and led me toward the solution with constant advice, support, and encouragement.
I would also like to thank the members of my committee, professors P. Fong, D. Radford,
F. Smith and S. R. Doty, for their interest in my work.
Thanks to Prof. P. Fong for the financial support through the last two terms of my study,
Prof. S. Smith and the UIC department of mathematics for the use of the SUN computer
network and to Andrew Mathas for his help with this system.
From August 1992 I was employed by the Loyola University in Chicago. I thank the De­
partment of Mathematical Sciences there for the support during that time.
To Ms L. Allen of the UIC department of mathematics I thank for typing Chapter 3 of this
thesis.
Part of the proofreading was done by my husband Michael Stirniman, to whom I owe
gratitude for many other reasons also.
EZ
iii
TABLE OF CONTENTS
CHAPTER PAGE
1 PRELIMINARIES AND MAIN THEOREMS 1
1.1 Introduction 1
1.2 Notation and Definitions 3
1.3 Unipotent classes in GF 5
1.4 The quadric Q 6
1.5 The main theorems 10
1.5.1 Theorem 1.1 10
1.5.2 Theorem 1.2 15
1.5.3 Theorem 1.3 22
1.5.4 Theorem 1.4 28
2 THE CASE OF ONE BLOCK 30
2.1 General remarks 30
2.2 A = ld 32
2.3 X = 2d 32
2.3.1 Discussion of a Frobenius map 34
2.4 X = rd,r>2 37
2.5 A closed formula for polynomials g^(q) 38
3 THE GENERAL CASE 41
3.1 Example 41
3.2 Some more notation 43
3.3 Compositions in 1-belt 47
3.4 Compositions in 2-belt 55
3.4.1 Discussion of a Frobenius map 56
3.5 Compositions in &-belt 62
3.6 The general case 71
3.6.1 Compositions in (l,0)-belt 71
3.6.2 Compositions in (2, l)-belt 72
3.6.3 Compositions in (k, k — ^)-belt 74
CITED LITERATURE 77
VITA 78
iv
SUMMARY
Let F be a finite field of q elements and let F be an algebraic closure of F , q odd. Let
q q
V be a 2m-dimensional vector space over F with a skew symmetric bilinear form [ , ]. Let
G be the group of linear transformations of V leaving [ , ] invariant, i.e., G is the symplectic
group Sp(2m,F). If P is a maximal parabolic subgroup of G, then X = G/P is a variety
isomorphic to the variety of totally isotropic subspaces of V of a given dimension k < m. For
a fixed unipotent element u of G, and a fixed unipotent element w of a Levi subgroup L of P,
and p the projection of F on I, we study the variety
X = { xP; u(xP) = xP and p{x_1ux) is conjugate to w in L }.
UiVJ
The number of F -rational points of this variety ( a polynomial in q) is calculated when L
q
is isomorphic to GL {F), i.e., k = m.
m
A closed formula is given for these polynomials in the case of one block. In the more general
case a recursive formula is given. Our calculations are done using the properties of symplectic
geometry.
The fact that these polynomials are useful in computing Green functions was the motivation
for the study of this variety.
v
CHAPTER 1
PRELIMINARIES AND MAIN THEOREMS
1.1 Introduction
Let F be a finite field of q elements and let F be an algebraic closure of F , q odd. Let V
q q
be a 2m-dimensional vector space over F with a nondegenerate skew symmetric bilinear form
[ , ]. Let G be the group of linear transformations of V leaving [ , ] invariant, i.e., G is the
symplectic group Sp(2m, F). If P is a maximal parabolic subgroup of G, then X = G/P is a
variety isomorphic to the variety of totally isotropic subspaces of V of a given dimension k < m.
Every maximal parabolic subgroup P of G has a Levi subgroup isomorphic to GLk x
Sp2{m-k)-
P can be regarded as the stabilizer of a fixed isotropic subspace of dimension k of V. Then
there is a one to one correspondence between the cosets xP of P and totally isotropic subspaces
W of dimension k of V. For a fixed unipotent element u of G, V becomes an .F[2]-module with
the action of t defined as t * v = (u — l)v. The partition of 2m corresponding to the Jordan
canonical form of u determines the type of the module V. Let a be a fixed unipotent element
of G. Let ttbea fixed unipotent element of a Levi subgroup L of P, and let p be the projection
of P on L. Then the variety
X = { xP ; u(xP) = xP and p(x 1ux) is conjugate to w in L }
u<w
is isomorphic to the variety of totally isotropic subspaces W as above, where each W is
fixed by u and w. The Jordan canonical form of w determines two partitions fi and v. Under
1