Computing the Discriminants of Brauer's Centralizer Algebras

This paper discusses a computational problem arising in the study of the structure theory of Brauer's orthogonal and symplectic centralizer algebras. The problem is to compute the ranks of certain combinatorially defined matrices Zm k(x) (these matrices are presented in §2). This computation is difficult because the sizes of the matrices Zm k(x) are enormous even for small values of m and k . However, there is a great deal of symmetry amongst the entries of the matrices. In this paper we show how to design algorithms that take full advantage of this symmetry, using the representation theory of the symmetric groups. We also present data collected using these algorithms and a number of conjectures about the centralizer algebras.