Invariant Theory for the free left-regular band and a q-analogue
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916888161812480 |
|---|---|
| author | Brauner, Sarah Commins, Patricia Reiner, Victor |
| author_facet | Brauner, Sarah Commins, Patricia Reiner, Victor |
| contents | We examine from an invariant theory viewpoint the monoid algebras for two monoids having large symmetry groups. The first monoid is the free left-regular band on $n$ letters, defined on the set of all injective words, that is, the words with at most one occurrence of each letter. This monoid carries the action of the symmetric group. The second monoid is one of its $q$-analogues, considered by K. Brown, carrying an action of the finite general linear group. In both cases, we show that the invariant subalgebras are semisimple commutative algebras, and characterize them using Stirling and $q$-Stirling numbers.
We then use results from the theory of random walks and random-to-top shuffling to decompose the entire monoid algebra into irreducibles, simultaneously as a module over the invariant ring and as a group representation. Our irreducible decompositions are described in terms of derangement symmetric functions introduced by Désarménien and Wachs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_11406 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Invariant Theory for the free left-regular band and a q-analogue Brauner, Sarah Commins, Patricia Reiner, Victor Combinatorics Representation Theory 05E10, 16W22, 60J10 We examine from an invariant theory viewpoint the monoid algebras for two monoids having large symmetry groups. The first monoid is the free left-regular band on $n$ letters, defined on the set of all injective words, that is, the words with at most one occurrence of each letter. This monoid carries the action of the symmetric group. The second monoid is one of its $q$-analogues, considered by K. Brown, carrying an action of the finite general linear group. In both cases, we show that the invariant subalgebras are semisimple commutative algebras, and characterize them using Stirling and $q$-Stirling numbers. We then use results from the theory of random walks and random-to-top shuffling to decompose the entire monoid algebra into irreducibles, simultaneously as a module over the invariant ring and as a group representation. Our irreducible decompositions are described in terms of derangement symmetric functions introduced by Désarménien and Wachs. |
| title | Invariant Theory for the free left-regular band and a q-analogue |
| topic | Combinatorics Representation Theory 05E10, 16W22, 60J10 |
| url | https://arxiv.org/abs/2206.11406 |