Howe duality of the symmetric group and a multiset partition algebra

TL;DR

This paper introduces the multiset partition algebra, a diagram algebra generalizing the partition algebra, and establishes its isomorphism with a symmetric group centralizer algebra, describing its representations and branching rules.

Contribution

It defines the multiset partition algebra and proves its isomorphism to a symmetric group centralizer algebra for certain parameters, expanding the understanding of algebraic structures related to symmetric groups.

Findings

The multiset partition algebra generalizes the partition algebra.

${ m Mf P}_{r,k}(x)$ is isomorphic to a symmetric group centralizer algebra for $x \\geq 2r$.

The paper describes the representations, branching rules, and restrictions of ${ m Mf P}_{r,k}(x)$.

Abstract

We introduce the multiset partition algebra, ${\rm M\!P}_{r,k}(x)$, 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 ${\rm M\!P}_{r,k}(x)$ is isomorphic to a centralizer algebra of the symmetric group, $S_n$, acting on the polynomial ring on the variables $x_{ij}$, $1\leq i \leq n$ and $1\leq j\leq k$. We describe the representations of ${\rm M\!P}_{r,k}(x)$, branching rule and restriction of its representations in the case that $x$ is an integer greater or equal to $2r$.