Super Multiset RSK and a Mixed Multiset Partition Algebra

TL;DR

This paper introduces the mixed multiset partition algebra and a generalized RSK algorithm to study representation theory related to tensor, symmetric, and exterior powers, revealing new algebraic decompositions.

Contribution

It extends the theory of partition algebras to exterior powers and generalizes the RSK algorithm for multisets, linking combinatorics with representation theory.

Findings

Enumerative results reflecting algebraic decompositions

Description of polynomial ring modules over GL and symmetric groups

Introduction of mixed multiset partition algebra

Abstract

Through dualities on representations on tensor powers and symmetric powers respectively, the partition algebra and multiset partition algebra have been used to study long-standing questions in the representation theory of the symmetric group. In this paper we extend this story to exterior powers, introducing the mixed multiset partition algebra as well as a generalization of the Robinson-Schensted-Knuth algorithm to two-row arrays of multisets with elements from two alphabets. From this algorithm, we obtain enumerative results which reflect representation-theoretic decompositions of this algebra. Furthermore, we use the generalized RSK algorithm to describe the decomposition of a polynomial ring in sets of commuting and anti-commuting variables as a module over both the general linear group and the symmetric group.