Topology and monoid representations II: Left regular bands of groups and Hsiao's monoid of ordered G-partitions

Abstract

The goal of this paper is to use topological methods to compute Ext between irreducible representations of von Neumann regular monoids in which Green's $L$- and $J$-relations coincide (e.g., left regular bands). Our results subsume those of Margolis et al. (2015) [22].

Applications include computing Ext between arbitrary simple modules and computing a quiver presentation for the algebra of Hsiao's monoid of ordered G-partitions (connected to the Mantaci-Reutenauer descent algebra for the wreath product $G\wr S_n$). We show that this algebra is Koszul, compute its Koszul dual and compute minimal projective resolutions of all the simple modules using topology. More generally, these results work for CW left regular bands of abelian groups. These results generalize the results of Margolis et al. (2021) [23].