Partition Algebras and the Invariant Theory of the Symmetric Group

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 $\varPhi _{k,n}: \mathsf {P}_k(n) \rightarrow \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})$ 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 $\varPhi _{k,n}$ has a nontrivial kernel which is generated as a two-sided ideal by a single idempotent. We describe the kernel and image of $\varPhi _{k,n}$ in terms of the orbit basis of $\mathsf {P}_k(n)$ and explain how the surjection $\varPhi _{k,n}$ can also be used to obtain the fundamental theorems of invariant theory for the symmetric group.

The second author gratefully acknowledges partial support from Simons Foundation grant 283311.