Howe duality of the symmetric group and a multiset partition algebra

Abstract

We introduce the multiset partition algebra, that has basis elements indexed by multiset partitions, where x is an indeterminate and r and k are non-negative integers. This algebra can be realized as a diagram algebra that generalizes the partition algebra. When x = n is an integer greater or equal to 2r, we show that is isomorphic to a centralizer algebra of the symmetric group, Sn , acting on the polynomial ring in the variables For the centralizer algebra we give a formula for the dimensions of the irreducible representations and a formula for the branching rule. We also show that the Kronecker coefficients occur as multiplicities in the restrictions of these irreducible modules.