Representations of certain generalized Schur algebras

This paper studies semigroup algebras A over a domain R for a certain family of semigroups which includes the symmetric groups, the full-transformation semigroups, and the rook semigroups. For each algebra A an A-module M on which A acts is introduced and an algebra B is defined as the commuting algebra of A acting on M. It is shown that A is also the commuting algebra of B acting on M. When A is a symmetric group algebra R[Sr], B is the corresponding Schur algebra SR(r,r). The representation theory of the semigroup algebras A is well known. This paper analyzes the irreducible representations of the B algebras. In many cases, including the symmetric group, full-transformation semigroup, and rook semigroup cases, a complete set of inequivalent irreducible representations of the B algebra is obtained. In particular, this gives what appears to be a new description of a complete set of irreducible representations for the Schur algebra SR(r,r).