Howe duality of the symmetric group and a multiset partition algebra

Abstract

We introduce the multiset partition algebra, ${\mathrm{M}\mathrm{P}}_{r,k}(x)$, that has bases elements indexed by multiset partitions, where $x$ is an indeterminate and $r$ and $k$ are non-negative integers. This algebra can be realized as a diagram algebra that generalizes the partition algebra. When $x$ is an integer greater or equal to $2r$, we show that ${\mathrm{M}\mathrm{P}}_{r,k}(x)$ is isomorphic to a centralizer algebra of the symmetric group, $S_n$, acting on the polynomial ring on the variables $x_{ij}$, $1\le i\le n$ and $1\le j\le k$. We describe the representations of ${\mathrm{M}\mathrm{P}}_{r,k}(x)$, branching rule and restriction of its representations in the case that $x$ is an integer greater or equal to $2r$.

24 pages