On subgroup adapted bases for representations of the symmetric group

TL;DR

This paper develops a novel analytical method to compute subduction coefficients and matrix representations for symmetric group representations, resolving multiplicity issues and applicable to large Young diagram differences.

Contribution

Introduces a new approach using a modified Schur-Weyl duality to analytically derive subduction coefficients and matrix representations, addressing multiplicity problems.

Findings

Analytical expressions for subduction coefficients obtained

Method resolves multiplicity issues in representation decomposition

Applicable to large row length differences in Young diagrams

Abstract

The split basis of an irreducible representation of the symmetric group, $S_{n+m}$, is the basis which is adapted to direct product subgroups of the form $S_{n} \times S_{m}$. In this article we have calculated symmetric group subduction coefficients relating the standard Young-Yamanouchi basis for the symmetric group to the split basis by means of a novel version of the Schur-Weyl duality. We have also directly obtained matrix representations in the split basis using these techniques. The advantages of this approach are that we obtain analytical expressions for the subduction coefficients and matrix representations and directly resolve issues of multiplicity in the subduction of $S_{n} \times S_{m}$ irreducible representations from $S_{n+m}$ irreducible representations. Our method is applicable to $S_{n+m}$ irreducible representations labelled by Young diagrams with large row length differences. We provide evidence for a systematic expansion in the inverse row length difference and tools to compute the leading contribution, which is exact when row length differences are infinite.