Left regular bands with symmetry

TL;DR

This paper explores the representation theory of invariant subalgebras of left regular band semigroup algebras under group actions, linking algebraic properties with combinatorial and topological structures, especially in geometric and CAT(0)-cube complex contexts.

Contribution

It characterizes when these invariant subalgebras are semisimple or commutative and provides topological formulas for their structure, extending previous work to new algebraic and geometric settings.

Findings

Invariant subalgebras can be semisimple or commutative under certain conditions.

Peirce components relate to the equivariant topology of intervals in the support semilattice.

Connections to Markov chains and generalizations of derangement representations are established.

Abstract

The representation theory of left regular band semigroup algebras is well-studied and known to have close connections with combinatorial topology, as established in the work of Margolis--Saliola--Steinberg ('15, '21). In this paper, we investigate the representation theory of the invariant subalgebras of left regular band semigroup algebras carrying the action of a finite group through the lens of group-equivariant combinatorial topology. We characterize when the invariant subalgebra is semisimple or commutative and examine the equivariant structure of the Peirce components of the semigroup algebra. For CW left regular bands, we interpret these Peirce components in terms of the equivariant topology of intervals in the support semilattice, yielding the Cartan invariants of the invariant subalgebras of left regular bands associated to CAT(0)-cube complexes. We also give a topological formula for the Peirce components for left regular bands with hereditary algebras. Finally, in specializing to left regular bands associated to geometric lattices, we explore generalizations of the Desarm\'{e}ni\'{e}n--Wachs derangement representation and their connections to Markov chains.