Left regular bands of groups and the Mantaci-Reutenauer algebra

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 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 they recover idempotents introduced by Vazirani. (c) 2023 Elsevier Inc. All rights reserved.