Finding splitting elements and maximal tori in matrix algebras

Abstract

Roughly speaking, a splitting element of a matrix algebra is an element with a maximal number of eigenvalues. Using randomization, splitting elements can be found very easily and have proved to be extremely useful in computational representation theory. Efficient deterministic algorithms for finding splitting elements have been known only for special cases. In this paper we present a deterministic polynomial time method for constructing splitting elements and closely related subalgebras – maximal tori – which works over a wide range of ground fields.