Partition Algebras and the Invariant Theory of the Symmetric Group

The symmetric group 𝖲_n and the partition algebra 𝖯_k(n) centralize one another in their actions on the k-fold tensor power 𝖬_n^⊗ k of the n-dimensional permutation module 𝖬_n of 𝖲_n. The duality afforded by the commuting actions determines an algebra homomorphism Φ_k,n: 𝖯_k(n) →𝖤𝗇𝖽_𝖲_n(𝖬_n^⊗ k) from the partition algebra to the centralizer algebra 𝖤𝗇𝖽_𝖲_n(𝖬_n^⊗ k), which is a surjection for all k, n ∈ℤ_≥ 1, and an isomorphism when n ≥ 2k. We present results that can be derived from the duality between 𝖲_n and 𝖯_k(n); for example, (i) expressions for the multiplicities of the irreducible 𝖲_n-summands of 𝖬_n^⊗ k, (ii) formulas for the dimensions of the irreducible modules for the centralizer algebra 𝖤𝗇𝖽_𝖲_n(𝖬_n^⊗ 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 𝖯_k(n). When 2k >n, the map Φ_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 Φ_k,n in terms of the orbit basis of 𝖯_k(n) and explain how the surjection Φ_k,n can also be used to obtain the fundamental theorems of invariant theory for the symmetric group.