Left Regular Bands of Groups and the Mantaci--Reutenauer algebra
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929712154017792 |
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| author | Bastidas, Jose Brauner, Sarah Saliola, Franco |
| author_facet | Bastidas, Jose Brauner, Sarah Saliola, Franco |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_14985 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Left Regular Bands of Groups and the Mantaci--Reutenauer algebra Bastidas, Jose Brauner, Sarah Saliola, Franco Combinatorics Representation Theory 05E10, 05E18, 16Gxx, 20M10 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. |
| title | Left Regular Bands of Groups and the Mantaci--Reutenauer algebra |
| topic | Combinatorics Representation Theory 05E10, 05E18, 16Gxx, 20M10 |
| url | https://arxiv.org/abs/2306.14985 |