The multiset partition algebra

We introduce the multiset partition algebra MPk(ξ) 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 MPk(n) and the symmetric group Sn on Symk(Fn). The construction of MPk(ξ) generalizes to any vector λ of non-negative integers yielding the algebra MPλ(ξ) over F[ξ] so that there is Schur—Weyl duality between the actions of MPλ(n) 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 MPk(n) and the generating function for the multiplicity of an irreducible polynomial representation of GLn(F) when restricted to Sn. We show that MPλ(ξ) embeds inside the partition algebra P|λ|(ξ). 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 MPλ(v) is a cellular algebra.