The Multiset Partition Algebra

TL;DR

This paper introduces the multiset partition algebra, establishes its Schur-Weyl duality with symmetric groups, and explores its structure, representations, and connections to other algebras, providing new tools for understanding symmetric functions and algebraic combinatorics.

Contribution

It generalizes the multiset partition algebra, proves its cellularity and semisimplicity under certain conditions, and develops an insertion algorithm for its module decomposition.

Findings

Establishes Schur-Weyl duality between $ ext{MP}_k(n)$ and $S_n$ actions.

Shows $ ext{MP}_ ext{lambda}( ext{ extbackslash xi})$ is a cellular algebra.

Provides generating functions for multiplicities of irreducible representations.

Abstract

We introduce the multiset partition algebra $\mathcal{MP}_k(\xi)$ over $F[\xi]$, where $F$ is a field of characteristic $0$ and $k$ is a positive integer. When $\xi$ is specialized to a positive integer $n$, we establish the Schur-Weyl duality between the actions of resulting algebra $\mathcal{MP}_k(n)$ and the symmetric group $S_n$ on $\text{Sym}^k(F^n)$. The construction of $\mathcal{MP}_k(\xi)$ generalizes to any vector $\lambda$ of non-negative integers yielding the algebra $\mathcal{MP}_{\lambda}(\xi)$ over $F[\xi]$ so that there is Schur-Weyl duality between the actions of $\mathcal{MP}_{\lambda}(n)$ and $S_n$ on $\text{Sym}^{\lambda}(F^n)$. We find the generating function for the multiplicity of each irreducible representation of $S_n$ in $\text{Sym}^\lambda(F^n)$, as $\lambda$ varies, in terms of a plethysm of Schur functions. As consequences we obtain an indexing set for the irreducible representations of $\mathcal{MP}_k(n)$, and the generating function for the multiplicity of an irreducible polynomial representation of $GL_n(F)$ when restricted to $S_n$. We show that $\mathcal{MP}_\lambda(\xi)$ embeds inside the partition algebra $\mathcal{P}_{|\lambda|}(\xi)$. Using this embedding, over $F$, we prove that $\mathcal{MP}_{\lambda}(\xi)$ is a cellular algebra, and $\mathcal{MP}_{\lambda}(\xi)$ is semisimple when $\xi$ is not an integer or $\xi$ is an integer such that $\xi\geq 2|\lambda|-1$. We give an insertion algorithm based on Robinson-Schensted-Knuth correspondence realizing the decomposition of $\mathcal{MP}_{\lambda}(n)$ as $\mathcal{MP}_{\lambda}(n)\times \mathcal{MP}_{\lambda}(n)$-module.