# Computing generators and relations for matrix algebras

Abstract (Summary)

We describe algorithms for computing a presentation for a matrix algebra over a
finite field, and for computing the basic algebra associated to such a matrix algebra.
We give correctness proofs of our algorithms, and implementations of them in the
Magma computer algebra system. We use these implementations to compute several
basic algebras.
Index words: Matrix Algebras, Finite Dimensional Algebras, Basic Algebras,
Generators and Relations, Morita Theory, Modular
Representation Theory.
Computing Generators and Relations for Matrix Algebras
by
Graham Y. Matthews
BSc Hons (First Class), The University of Auckland, 1989
MSc (With Distinction), The University of Auckland, 1991
Graduate Diploma, The Australian National University, 1997
A Dissertation Submitted to the Graduate Faculty
of The University of Georgia in Partial Fulfillment
of the
Requirements for the Degree
Doctor of Philosophy
Athens, Georgia
2004
c? 2004
Graham Y. Matthews
All Rights Reserved
Computing Generators and Relations for Matrix Algebras
by
Graham Y. Matthews
Approved:
Major Professor:
Committee:
Jon Carlson
Brian Boe
Leonard Chastkofsky
Elham Izadi
Robert Varley
Electronic Version Approved:
Maureen Grasso
Dean of the Graduate School
The University of Georgia
August 2004
Preface
Modular representation theory is the study of the realizations of an algebra A over a
field K of characteristic p, as a subalgebra of the endomorphisms of some K-vector
space V . In computational modular representation theory we usually take K to be
finite, and both V and A to be finite dimensional over K, so that endomorphisms of V
can be represented as n×n matrices over K, where n is the K-dimension of V . In the
computational setting A is usually implicitly defined via a sequence ? = {?1, . . . , ?t}
of n × n matrices over K, so A is the subalgebra of Mn(K) generated by ?.
Two natural questions arise. First, can we compute a presentation for A in terms
of generators and relations, and if so, can this be done in a somewhat canonical
way? The more canonical the presentation, the more useful it becomes in answering
related questions, such as whether two algebras are isomorphic. Second, can we
compute the basic algebra associated to A? The basic algebra, B, is a usually much
smaller dimensional algebra, with the property that A and B are Morita equivalent,
i.e., the module category for A and the module category for B are categorically
equivalent, and hence representations of A and of B are ‘essentially the same’.
Previous approaches to these questions have mainly focussed on the second
problem, and have assumed that A is the group algebra of some finite group G.
The techniques employed center on finding idempotents e ? KG, usually via special
subgroups of G, such that the condensed algebra eKGe is Morita equivalent to the
group algebra KG. There is a large body of work [16] on how to both construct and
recognize such idempotents. The essential problem with this approach is that there is
no guarantee that if g1, . . . , gn generate KG, then eg1e, . . . , egne will generate eKGe.
iv
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This dissertation attempts to answer both problems via a somewhat different
approach. We still attempt to find idempotents in A, and then condense A with
respect to these idempotents, but we do not assume that A is the group algebra
of some finite group. Rather we work directly with A as a subalgebra of a matrix
algebra. We also treat the two problems as being intimately related – our solution
to the second problem yields a natural solution to the first.
We start by proving that A can be ‘split’ as the direct sum of a subalgebra A?
isomorphic (as an algebra) to A modulo its Jacobson radical J(A), and the twosided
ideal J(A). Next we show how to find generators and relations for A?. These
generators are constructed in a canonical way, with two generators and a family of
four relations per simple A-module. One of the two generators is actually in the
basic algebra B for A, and the other becomes zero when we condense to form B. We
then show how to construct a generating set for J(A) as a two-sided ideal. While this
generating set is not quite canonical, it has the interesting property that it is wholly
contained in B. Hence the set not only generates J(A), but also J(B). Our careful
construction of generators for A? and J(A) via elements of A that are either in B,
or condense to zero within B, allows us to both construct B as a matrix algebra,
and to compute a presentation for B via generators and relations. We conclude
by computing the basic algebra associated to several algebras, including the group
algebra of the Mathieu group M11 in characteristic 2.
We provide algorithms for all our constructions, along with proofs of their correctness.
Appendix A contains implementations of our algorithms in the Magma
computer algebra system
Bibliographical Information:

Advisor:

School:The University of Georgia

School Location:USA - Georgia

Source Type:Master's Thesis

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