Howe duality of the symmetric group and a multiset partition algebra

Abstract

We introduce the multiset partition algebra, MP r,k (x), that has bases 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 is an integer greater or equal to 2r, we show that MP r,k (x) is isomorphic to a centralizer algebra of the symmetric group, Sn, acting on the polynomial ring on the variables xij , 1 ≤ i ≤ n and 1 ≤ j ≤ k. We describe the representations of MP r,k (x), branching rule and restriction of its representations in the case that x is an integer greater or equal to 2r.

Stability theorems for multiplicities in graded $S_n$-modules

“…One key aspect of the proof of this stabilization result is the following result of the first named author of this paper (see [7]). The total degree version found in Proposition 5.2 of [10] can also be used to provide an alternate proof. Considering…”

A Combinatorial Model for the Decomposition of Multivariate Polynomial Rings as $S_n$-Modules

“…An important application of the main theorem of this paper and the double centralizer theorem [CR, GW] is to give an interpretation to the dimensions of the irreducible representations of the centralizer of CS n when it acts on multivariate polynomial rings (see [NPS,OZ3]).…”