A survey of semisimple algebras in algebraic combinatorics

This is a survey of semisimple algebras of current interest in algebraic combinatorics, with a focus on questions which we feel will be new and interesting to experts in group algebras, integral representation theory, and computational algebra. The algebras arise primarily in two families: coherent algebras and subconstituent (aka. Terwilliger) algebras. Coherent algebras are subalgebras of full matrix algebras having a basis of 01-matrices satisfying the conditions that it be transpose-closed, sum to the all 1's matrix, and contain a subset Delta that sums to the identity matrix. The special case when Delta is a singleton is the important case of an adjacency algebra of a finite association scheme. A Terwilliger algebra is a semisimple extension of a coherent algebra by a set of diagonal 01-matrices determined canonically from its basis elements and a choice of row. We will survey the current state of knowledge of the complex, real, rational, modular, and integral representation theory of these semisimple algebras, indicate their connections with other areas of mathematics, and present several open questions.