The multiset partition algebra

Abstract

We introduce the multiset partition algebra ${\cal M}{{\cal P}_k}\left(\xi \right)$ over the polynomial ring F[ξ], where F is a field of characteristic 0 and k is a positive integer. When ξ is specialized to a positive integer n, we establish the Schur—Weyl duality between the actions of resulting algebra ${\cal M}{{\cal P}_k}\left(n \right)$ and the symmetric group S_n on Sym^k(Fⁿ). The construction of ${\cal M}{{\cal P}_k}\left(\xi \right)$ generalizes to any vector λ of non-negative integers yielding the algebra ${\cal M}{{\cal P}_\lambda}\left(\xi \right)$ over F[ξ] so that there is Schur—Weyl duality between the actions of ${\cal M}{{\cal P}_\lambda}\left(n \right)$ and S_n on Sym^λ(Fⁿ). We find the generating function for the multiplicity of each irreducible representation of S_n in Sym^λ(Fⁿ), as λ varies, in terms of a plethysm of Schur functions. As consequences we obtain an indexing set for the irreducible representations of ${\cal M}{{\cal P}_k}\left(n \right)$ and the generating function for the multiplicity of an irreducible polynomial representation of GL_n(F) when restricted to S_n. We show that ${\cal M}{{\cal P}_\lambda}\left(\xi \right)$ embeds inside the partition algebra ${{\cal P}_{\left| \lambda \right|}}\left(\xi \right)$. Using this embedding, we show that the multiset partition algebras are generically semisimple over F. Also, for the specialization of ξ at v in F, we prove that ${\cal M}{{\cal P}_\lambda}\left(v \right)$ is a cellular algebra.