Branching from the General Linear Group to the Symmetric Group and the Principal Embedding

TL;DR

This paper investigates the branching rules for how finite-dimensional representations of GL(n,C) decompose into irreducible symmetric group representations upon restriction, building on prior work on embeddings of sl_2 in sl_n.

Contribution

It characterizes which symmetric group irreducibles appear with positive multiplicity in the restriction of GL(n,C) representations, extending previous results on principal embeddings.

Findings

Identifies conditions for positive multiplicity of symmetric group irreducibles

Provides explicit branching rules for GL(n,C) to symmetric groups

Builds on prior bounds for sl_2 embeddings in sl_n

Abstract

Let S be a principally embedded sl_2 subalgebra in sl_n for n > 2. A special case of results of the third author and Gregg Zuckerman implies that there exists a positive integer b(n) such that for any finite-dimensional irreducible sl_n representation, V, there exists an irreducible S-representation embedding in V with dimension at most b(n). In a 2017 paper (joint with Hassan Lhou), they prove that b(n)=n is the sharpest possible bound, and also address embeddings other than the principal one. These results concerning embeddings may by interpreted as statements about plethysm. Then, a well known result about these plethysms can be interpreted as a "branching rule". Specifically, a (finite dimensional) representation of GL(n,C) will decompose into irreducible representations of the symmetric group when it is restricted to the subgroup consisting of permutation matrices. The question of which irreducible representations of the symmetric group occur with positive multiplicity is the topic of this paper, applying the previous work of Lhou, Zuckerman, and the third author.