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 Sn on Symk(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 Sn on Symλ(Fn). We find the generating function for the multiplicity of each irreducible representation of Sn in Symλ(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 GLn(F) when restricted to Sn. 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.