Computing generators and relations for matrix algebras

by (Graham Yakov), 1967- Matthews

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 v 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