Left regular bands with symmetry

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.