Dispersing representations of semi-simple subalgebras of complex matrices

Abstract

In this paper we consider the problem of determining the maximum dimension of P?(A!B)P, where A and B are unital, semi-simple subalgebras of the set Mn of n⇥n complex matrices, and P 2 M2n is a projection of rank n. We exhibit a number of equivalent formulations of this problem, including the one which occupies the majority of the paper, namely: determine the minimum dimension of the space A\ S−1BS, where S is allowed to range over the invertible group GL(n,C) of Mn. This problem in turn is seen to be equivalent to the problem of finding two automorphisms ↵ and " of Mn for which the dimension of ↵(A)+"(B) is maximised. It is this phenomenon which gives rise to the title of the paper.