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Let $V$ be an $n$-dimensional vector space over some commutative ring $R$.The symmetric group $\mathfrak S_n$ acts on tensor space $V^{\otimes r}$ byrestricting the natural action of $GL(V)$ on tensor space to its subgroup$\mathfrak S_n$. We construct an action of the corresponding Iwahori-Heckealgebra $\mathcal H_{R,q}(\mathfrak S_n)$ which specializes to the action of$\mathfrak S_n$, if $q$ is taken to $1$. The centralizing algebra of thisaction 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 byHalverson and Thiem by different means a few years ago.

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