Left Regular Bands of Groups and the Mantaci--Reutenauer algebra
Jose Bastidas, Sarah Brauner, Franco Saliola
TL;DR
The paper develops a comprehensive idempotent theory for left regular bands of groups (LRBGs), unifying the idempotent frameworks of left regular bands and finite group algebras. It constructs complete systems of primitive orthogonal idempotents (CSoPOI) for LRBG algebras by combining LRB-support techniques with group-algebra idempotents, and extends these to LRBGs with symmetry, yielding invariant-subalgebra idempotents. A central thread is Hsiao’s LRBGs ${\Sigma_n[G]}$, which connect to the Mantaci–Reutenauer algebras ${\sf MR}_n[G]$; for abelian $G$, the authors provide explicit bases, change-of-basis formulas, and closed-form CSoPOI for ${\sf MR}_n[G]$, recovering Vazirani’s idempotents when $G\cong C_2$. The framework generalizes to LRBaGs and yields a systematic pathway to study MR algebras and their representations, including explicit idempotents and their connections to descent-like algebras. Overall, the work integrates semigroup theory, representation theory, and combinatorial Hopf-algebra structures to illuminate the structure and representation theory of LRBGs and their invariants.
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.