The quiver of an algebra associated to the Mantaci-Reutenauer descent algebra and the homology of regular semigroups

TL;DR

This paper develops the homology theory for algebras of regular semigroups, characterizes directedness, and computes the quiver of related algebras, linking semigroup theory with representation theory of wreath products and descent algebras.

Contribution

It introduces a homology framework for regular semigroup algebras and computes their quivers, extending previous results to new algebraic structures.

Findings

Homology theory for regular semigroup algebras developed

Directedness characterized for these algebras

Quiver of Hsiao's algebra computed, related to Mantaci-Reutenauer descent algebra

Abstract

We develop the homology theory of the algebra of a regular semigroup, which is a particularly nice case of a quasi-hereditary algebra in good characteristic. Directedness is characterized for these algebras, generalizing the case of semisimple algebras studied by Munn and Ponizovksy. We then apply homological methods to compute (modulo group theory) the quiver of a right regular band of groups, generalizing Saliola's results for a right regular band. Right regular bands of groups come up in the representation theory of wreath products with symmetric groups in much the same way that right regular bands appear in the representation theory of finite Coxeter groups via the Solomon-Tits algebra of its Coxeter complex. In particular, we compute the quiver of Hsiao's algebra, which is related to the Mantaci-Reutenauer descent algebra.