Iwahori-Hecke algebras acting on tensor space by $q$-deformed letter permutations and $q$-partition algebras (2311.03156v3)

Abstract: 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.