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Characteristic-Free Representation Theory of GLn{Z):
Some Homological Aspects
A Dissertation
Presented To
The Faculty of the Graduate School of Arts and Sciences
Brandeis University
Department of Mathematics
Professor David Buchsbaum, Advisor
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
by
Upendra B. Kulkami
February 1999
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UMI Number: 9917919
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This dissertation, directed and approved by Upendra B. Kulkami’s Committee, has been
accepted and approved by the faculty of Brandeis University in partial fulfillment of the
requirements for the degree of:
DOCTOR OF PHILOSOPHY
Dean of Arts and Sciences
Dissertation Committee
David Buchsbaum (Chair), Ph.D.
Paul Monsky, Ph.D.
vy
Kaan Akin, Ph.D.
Department of Mathematics
University of Oklahoma
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Dedicated to the memory of my father
Silent waters run deep.
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ACKNOWLEDGEMENTS
David Buchsbaum believed in me and inspired me. For this, for his patience and
generosity, and for suggesting the problem that wandered into the investigations
presented here, I am very grateful.
It is a pleasure to thank Kari Vilonen and Alexei Rudakov for sharing with me
much beautiful mathematics and for showing me how not to be afraid of it.
I am grateful to everyone who made the department at Brandeis such a pleasant
home for my mathematical growth. I thank Maurice Auslander, Ira Gessel, Gerry
Schwarz, other professors, many fellow students and certainly earlier mathematics
teachers for enriching me; the departmental staff—Janet, Trish and Beth—for
their efficient help in all things practical. A special thank you to Susan Parker for
advice and tremendous support in the art of teaching and in other matters.
Many people lent a patient yet engaged ear to parts of this thesis at various
stages: K. Akin, V. Deodhar, D. Flores, M. Klucznik, A. Rudakov, R. Sanchez,
K. Vilonen, J. Weyman and of course, that ever encouraging audience of one,
David Buchsbaum; I appreciate their interest very much. K. Akin also made
useful and detailed comments on the previous version of this thesis.
If the last six checkered years of personal growth have left many fond memories, it
is because of several special people at Brandeis and elsewhere. The ties of family
and friendships, old and newer, have, perhaps unbeknownst to those involved,
sustained and at the same time shaped me in no small measure. These debts of
the past flowing into the present are deeply, if silently, cherished.
iv
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ABSTRACT
Characteristic-Free Representation Theory of GLn(Z):
Some Homological Aspects
A dissertation presented to the Faculty of the
Graduate School of Arts and Sciences of Brandeis
University, Waltham, Massachusetts
by Upendra B. Kulkarni
We present a simple recursive algorithm to compute the alternating product of
the cardinalities of Ext groups between Weyl modules for GLn(Z). For a class
of examples, we compute all the Ext groups. These results axe based on a new
method to reduce certain problems about polynomial representations of one degree
to problems in smaller degree. The reduction uses a result of independent interest:
we show the representative of an interesting functor involving tensor product to
be a characteristic-free module corresponding to a skew partition. On the level of
characters, this result is analogous to the following adjoint formula for the scalar
product on symmetric functions: (s\ , sM/) = (sx/M, /).
v
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TABLE OF CONTENTS
Introduction 1
0. Background and Notation 5
1. Generalities about Ext Groups 9
2. Construction of the Representative 15
3. Applications to the Extensions between Weyl Modules 23
4. Possible Directions for Further Work 40
References 43
vi
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INTRODUCTION
The rational representations of a reductive algebraic group G over a field of char
acteristic 0 have the following nice properties: 1) They are completely reducible.
2) The irreducibles are constructed as follows: let A range over dominant inte
gral weights with respect to a fixed maximal torus and a fixed Borel subgroup
containing it. Then the irreducibles are just the induced modules IndL%{A) or
their contravariant duals K\, called the Weyl modules. However once we work
over some other base, say an algebraically closed field of characteristic p > 0,
both these assertions fail. Thus 1) Nontrivial Ext groups appear and 2) The irre
ducibles are still indexed by A, but induced and Weyl modules are not necessarily
irreducible. Even though the exact nature of irreducibles M\ is intimately tied to
the characteristic, they are always obtained as the unique irreducible submodules
of Indg(A) as well as quotients of K\. Therefore knowledge of induced mod
ules and Weyl modules is valuable to understand the representation category. In
this thesis we study some aspects of the homological behaviour of these modules
for the group GLn. Much of the general set-up extends easily to other classical
groups as well, but several key obstacles need to be overcome to obtain similar
explicit results. We also note that the translation from GLn to SLn is immediate
(as usual). It is more convenient for us to work with GLn, essentially because
the comultiplication on its coordinate ring (induced by the group multiplication)
1
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