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

Geetha Thangavelu, Richard Dipper

Published 2023-11-06Version 1

Let $V$ be an $n$-dimensional vector space over some commutative ring $R$. The symmetric group $\mathfrak S_n$ acts on tensor space $V^{\otimes r}$ by restricting the natural action of $GL(V)$ on tensor space to its subgroup $\mathfrak S_n$. We construct an action of the corresponding Iwahori-Hecke algebra $\mathcal H_{R,q}(\mathfrak S_n)$ which specializes to the action of $\mathfrak S_n$, if $q$ is taken to $1$. The centralizing algebra of this action is called $q$-partition algebra $\mathcal P_{R,q}(n,r)$. 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.