Cohomology of coloured partition algebras

TL;DR

This paper proves that coloured partition algebras exhibit homological stability, with their homology groups becoming isomorphic to those of a wreath product, extending previous results for standard partition algebras.

Contribution

It demonstrates homological stability for coloured partition algebras, generalizing earlier work on standard partition algebras and connecting to wreath product homology.

Findings

Homology groups of coloured partition algebras are stably isomorphic to wreath product homology.

Generalization of Boyd--Hepworth--Patzt and Boyde's results to coloured partition algebras.

Establishment of homological stability for these algebras.

Abstract

Coloured partition algebras were introduced by Bloss and exhibit a Schur-Weyl duality with certain complex reflection groups. In this paper we show that these algebras exhibit homological stability by demonstrating that their homology groups are stably isomorphic to the homology groups of a wreath product, generalizing work of Boyd--Hepworth--Patzt and Boyde for the usual partition algebras.