University of Sydney Algebra Seminar

James East (UWS)

Friday 26 September, 12:05-12:55pm, Place: 373

Enumeration of idempotents in partition monoids

Partition algebras arise in representation theory and statistical mechanics. These algebras (which contain the Brauer and Temperley-Lieb algebras) have bases consisting of certain diagrams, with the product of two basis elements always being a scalar multiple of another basis element. As such, they may also be constructed as twisted semigroup algebras of the partition monoids, a natural class of diagram semigroups that canonically embed the full transformation semigroups, symmetric inverse semigroups, and more. Although many results exist regarding the idempotents of these diagram semigroups (for example, every proper ideal is idempotent generated, and the rank and idempotent rank of these ideals are known), the number of idempotents has remained unknown for quite some time. In this talk I'll report on joint work with Igor Dolinka, Athanasios Evangelou, Des FitzGerald, Nick Ham, James Hyde and Nick Loughlin on the enumeration of idempotent partitions.