Left Regular Bands of Groups and the Mantaci--Reutenauer algebra

TL;DR

This paper develops an idempotent theory for left regular bands of groups (LRBGs), generalizing group and band algebras, and applies it to explicitly construct primitive orthogonal idempotents in the Mantaci--Reutenauer algebra.

Contribution

It introduces a systematic approach to study LRBGs and constructs explicit idempotents in the Mantaci--Reutenauer algebra for any finite group.

Findings

Constructed complete systems of primitive orthogonal idempotents in ${ m MR}_n[G]$

Derived closed form expressions for idempotents when G is abelian

Recovered Vazirani's idempotents for G as cyclic group of order two

Abstract

We develop the idempotent theory for algebras over a class of semigroups called left regular bands of groups (LRBGs), which simultaneously generalize group algebras of finite groups and left regular band (LRB) algebras. Our techniques weave together the representation theory of finite groups and LRBs, opening the door for a systematic study of LRBGs in an analogous way to LRBs. We apply our results to construct complete systems of primitive orthogonal idempotents in the Mantaci--Reutenauer algebra ${\sf{MR}}_n[G]$ associated to any finite group $G$. When $G$ is abelian, we give closed form expressions for these idempotents, and when $G$ is the cyclic group of order two, we prove that these recover idempotents introduced by Vazirani.