Partition Algebras and the Invariant Theory of the Symmetric Group

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.