Partition Algebras and the Invariant Theory of the Symmetric Group

TL;DR

This paper explores the duality between the symmetric group and partition algebra, revealing new formulas for multiplicities, dimensions, and character values, and connecting combinatorial objects with invariant theory.

Contribution

It provides a detailed analysis of the duality between the symmetric group and partition algebra, including explicit descriptions of kernels, images, and applications to invariant theory.

Findings

Formulas for irreducible module multiplicities

Dimensions of centralizer algebra modules

Bijection between vacillating and set-partition tableaux

Abstract

The symmetric group $\mathsf{S}_n$ and the partition algebra $\mathsf{P}_k(n)$ centralize one another in their actions on the $k$-fold tensor power $\mathsf{M}_n^{\otimes k}$ of the $n$-dimensional permutation module $\mathsf{M}_n$ of $\mathsf{S}_n$. The duality afforded by the commuting actions determines an algebra homomorphism $\Phi_{k,n}: \mathsf{P}_k(n) \to \mathsf{End}_{\mathsf{S}_n}(\mathsf{M}_n^{\otimes k})$ from the partition algebra to the centralizer algebra $\mathsf{End}_{\mathsf{S}_n}(\mathsf{M}_n^{\otimes k})$, which is a surjection for all $k, n \in \mathbb{Z}_{\ge 1}$, and an isomorphism when $n \ge 2k$. We present results that can be derived from the duality between $\mathsf{S}_n$ and $\mathsf{P}_k(n)$; for example, (i) expressions for the multiplicities of the irreducible $\mathsf{S}_n$-summands of $\mathsf{M}_n^{\otimes k}$, (ii) formulas for the dimensions of the irreducible modules for the centralizer algebra $\mathsf{End}_{\mathsf{S}_n}(\mathsf{M}_n^{\otimes k})$, (iii) a bijection between vacillating tableaux and set-partition tableaux, (iv) identities relating Stirling numbers of the second kind and the number of fixed points of permutations, and (v) character values for the partition algebra $\mathsf{P}_k(n)$. When $2k >n$, the map $\Phi_{k,n}$ has a nontrivial kernel which is generated as a two-sided ideal by a single idempotent. We describe the kernel and image of $\Phi_{k,n}$ in terms of the orbit basis of $\mathsf{P}_k(n)$ and explain how the surjection $\Phi_{k,n}$ can also be used to obtain the fundamental theorems of invariant theory for the symmetric group.