THE MULTISET PARTITION ALGEBRA

We introduce the multiset partition algebra MPk(xi) over the polynomial ring F [xi], where F is a field of characteristic 0 and k is a positive integer. When xi 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 S-n on Sym(k)(F-n). The construction of MPk(xi) generalizes to any vector lambda of non-negative integers yielding the algebra MP lambda (xi) over F[xi] so that there is Schur-Weyl duality between the actions of MP lambda (n) and S-n on Sym(lambda)(F-n). We find the generating function for the multiplicity of each irreducible representation of S-n in Sym(lambda) (F-n), as lambda 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 GL(n)(F) when restricted to Sn. We show that MP lambda(xi) embeds inside the partition algebra P-|lambda|(xi). Using this embedding, we show that the multiset partition algebras are generically semisimple over F. Also, for the specialization of xi at v in F, we prove that MP lambda (v) is a cellular algebra.