Commutative Centralizer Algebras and Related Structures

Abstract

Let M (x) be a completely reducible matrix representation of a group Gover a field K. The set C (M) of matrices T over K such that TM (x) = M (x)T for all x E G forms an algebra over K called the centralizer algebra of M. In this paper we investigate the centralizer algebras of permutation representations. We prove a sufficiency condition for the commutativity of centralizer algebras of semi-direct products and also obtain several commutativity results coming from some classes of finite groups using character theory and other techniques. Finally we show how these algebras arise in various contexts and discuss some related structures.

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