Cellularity of inverse semigroup algebras

Speaker: James East, University of Sydney

Abstract:The study of cellular algebras was initiated by Graham and Lehrer in 1996, the theory of which provides a unified framework for understanding the representation theory of several important algebras. There are many parallels between the structure of cellular algebras (which involves multiplicative properties of bases, involutions, and partially ordered sets) and the structure of finite inverse semigroups (inherited from Green's relations). It turns out that the semigroup algebra of a finite inverse semigroup is cellular if (i) the group algebras of its maximal subgroups have cellular group algebras, and (ii) a natural compatibility assumption is satisfied. We will look at several examples including the symmetric inverse semigroup (known to representation theorists as ``the rook monoid''), the dual symmetric inverse semigroup, the monoid of order-preserving partial permutations, and semilattices of groups.