Howe duality of the symmetric group and a multiset partition algebra

2020/07/14 by Orellana, Rosa, Zabrocki, Mike
#05E10 (Primary) 05E05 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2007.07370

Abstract

We introduce the multiset partition algebra, \rm M P_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 \rm M P_r,k(x) is isomorphic to a centralizer algebra of the symmetric group, S_n, acting on the polynomial ring on the variables x_ij, 1≤ i ≤ n and 1≤ j≤ k. We describe the representations of \rm M P_r,k(x), branching rule and restriction of its representations in the case that x is an integer greater or equal to 2r.

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