Iwahori-Hecke algebras acting on tensor space by $q$-deformed letter permutations and $q$-partition algebras
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916296660090880 |
|---|---|
| author | Thangavelu, Geetha Dipper, Richard |
| author_facet | Thangavelu, Geetha Dipper, Richard |
| contents | Let $R$ be a commutative ring with identity and let $V$ be a free $R$-module of rank $n$ for some $n\in\mathbb{N}$. Fixing an $R$-basis $\mathcal{E}$ of $V$, the symmetric group $\mathfrak{S}_n$ acts on $V$ by permuting $\mathcal{E}$ and hence on tensor space $V^{\otimes r}$ for $r\in\mathbb{N}$ via the usual tensor product action turning $V$ and $V^{\otimes r}$ into $R\mathfrak{S}_n$-modules. For units $q$ in $R$ we construct an action of the corresponding Iwahori-Hecke algebra $\mathcal{H}_{R,q}(\mathfrak{S}_n)$ which specializes to the action of $R\mathfrak{S}_n$, if $q$ is taken to $1$. The centralizing algebra of this action is called the $q$-partition algebra $\mathcal{P}_{R,q}(n,r)$. Let $R$ be a field of characteristic not dividing $q$. We prove, that $\mathcal{P}_{R,q}(n,r)$ is isomorphic to the $q$-partition algebra defined by Halverson and Thiem by different means a few years ago. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_03156 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Iwahori-Hecke algebras acting on tensor space by $q$-deformed letter permutations and $q$-partition algebras Thangavelu, Geetha Dipper, Richard Representation Theory 20C15, 20D15(Primary), 20C33, 20D20 (Secondary) Let $R$ be a commutative ring with identity and let $V$ be a free $R$-module of rank $n$ for some $n\in\mathbb{N}$. Fixing an $R$-basis $\mathcal{E}$ of $V$, the symmetric group $\mathfrak{S}_n$ acts on $V$ by permuting $\mathcal{E}$ and hence on tensor space $V^{\otimes r}$ for $r\in\mathbb{N}$ via the usual tensor product action turning $V$ and $V^{\otimes r}$ into $R\mathfrak{S}_n$-modules. For units $q$ in $R$ we construct an action of the corresponding Iwahori-Hecke algebra $\mathcal{H}_{R,q}(\mathfrak{S}_n)$ which specializes to the action of $R\mathfrak{S}_n$, if $q$ is taken to $1$. The centralizing algebra of this action is called the $q$-partition algebra $\mathcal{P}_{R,q}(n,r)$. Let $R$ be a field of characteristic not dividing $q$. We prove, that $\mathcal{P}_{R,q}(n,r)$ is isomorphic to the $q$-partition algebra defined by Halverson and Thiem by different means a few years ago. |
| title | Iwahori-Hecke algebras acting on tensor space by $q$-deformed letter permutations and $q$-partition algebras |
| topic | Representation Theory 20C15, 20D15(Primary), 20C33, 20D20 (Secondary) |
| url | https://arxiv.org/abs/2311.03156 |