The algebra of G-relations

Abstract

In this paper, we study a tower {A^G_n(d):n≥1} of finite-dimensional algebras; here, G represents an arbitrary finite group, d denotes a complex parameter, and the algebra A^G_n (d) has a basis indexed by 'G-stable equivalence relations' on a set where G acts freely and has n orbits. We show that the algebra A^G_n (d) is semi-simple for all but a finite set of values of d, and determine the representation theory (or, equivalently, the decomposition into simple summands) of this algebra in the 'generic case'. Finally we determine the Bratteli diagram of the tower {A^G_n(d):n≥1} (in the generic case).