The Multiset Partition Algebra

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