Mِbius functions and semigroup representation theory

This paper explores several applications of Mِbius functions to the representation theory of finite semigroups. We extend Solomonʹs approach to the semigroup algebra of a finite semilattice via Mِbius functions to arbitrary finite inverse semigroups. This allows us to explicitly calculate the orthogonal central idempotents decomposing an inverse semigroup algebra into a direct product of matrix algebras over group rings. We also extend work of Bidigare, Hanlon, Rockmore and Brown on calculating eigenvalues of random walks associated to certain classes of finite semigroups; again Mِbius functions play an important role.