Tableau Correspondences and Representation Theory

TL;DR

This paper explores how combinatorial tableau bijections can be used to derive various fundamental decompositions and dualities in the representation theory of general linear and symmetric groups.

Contribution

It introduces new combinatorial methods to obtain classical representation decompositions, including Howe's dualities and Gelfand models, from tableau correspondences.

Findings

Derived Gelfand models using tableaux

Revealed new combinatorial proofs of Schur-Weyl decomposition

Established multiplicity-free decompositions indexed by threshold partitions

Abstract

We deduce decompositions of natural representations of general linear groups and symmetric groups from combinatorial bijections involving tableaux. These include some of Howe's dualities, Gelfand models, the Schur-Weyl decomposition of tensor space, and multiplicity-free decompositions indexed by threshold partitions.