Invariant Theory for the free left-regular band and a q-analogue

Abstract

We examine from an invariant theory viewpoint the monoid algebras for two monoids having large symmetry groups:• the free left-regular band on n letters, acted on by the symmetric group, and • one of its q-analogues, acted on by the finite general linear group. In both cases, we show that the invariant subalgebras are semisimple commutative algebras, and characterize them using Stirling and q-Stirling numbers.We then use results from the theory of random walks and random-to-top shuffling to decompose the entire monoid algebra into irreducibles, simultaneously as a module over the invariant ring and as a group representation. Our irreducible decompositions are described in terms of derangement symmetric functions introduced by Désarménien and Wachs.