Partition Algebras and the Invariant Theory of the Symmetric Group

Abstract

The symmetric group Sn and the partition algebra Pk(n) centralize one another in their actions on the k-fold tensor power (forumala presented). of the n-dimensional permutation module Mn of Sn. The duality afforded by the commuting actions determines an algebra homomorphism (forumala presented). from the partition algebra to the centralizer algebra (forumala presented)., which is a surjection for all (forumala presented)., and an isomorphism when (forumala presented).. We present results that can be derived from the duality between (forumala presented). and (forumala presented). for example, (i) expressions for the multiplicities of the irreducible (forumala presented). -summands of (forumala presented)., (ii) formulas for the dimensions of the irreducible modules for the centralizer algebra (forumala presented)., (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 Pk(n). When (forumala presented)., the map (forumala presented). has a nontrivial kernel which is generated as a two-sided ideal by a single idempotent. We describe the kernel and image of (forumala presented). in terms of the orbit basis of (forumala presented). and explain how the surjection (forumala presented). can also be used to obtain the fundamental theorems of invariant theory for the symmetric group.