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

Alexander Heaton,Jeb F Willenbring,Songpon Sriwongsa

doi:10.5802/alco.138

Abstract

Let S be a principally embedded 𝔰𝔩 2 -subalgebra in 𝔰𝔩 n for n≥3. 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.

Highlights

These results concerning embeddings may be interpreted as statements about plethysm
In this paper we show a relationship between the principal embedding and branching from GLn to the symmetric group
Any work related to branching rules has benefited from the older extensive work of R

Summary

An algorithm for branching

Branching from GLn to Sn is among the class of problems which have an algorithm we can use to find the answer in any specific (finite) case, but lacks a general description, formula, or combinatorial explanation. Diagonalizable elements are dense in GLn and so we know the character of an irreducible representation of the general linear group is given by evaluating a Schur function in n variables corresponding to parameters of the maximal torus inside GLn. Diagonalizable elements are dense in GLn and so we know the character of an irreducible representation of the general linear group is given by evaluating a Schur function in n variables corresponding to parameters of the maximal torus inside GLn Replacing these variables with the corresponding eigenvalues (of correct multiplicities) for a permutation matrix of each cycle type, we create the trace of the operator of an element of the symmetric group acting on that same vector space (the representation of GLn whose character we have taken). By doing this over all possible cycle types, we find the character viewed as a representation of the symmetric group. By taking the inner product with irreducible characters of the symmetric group we can find the multiplicities of each irreducible representation of Sn inside the original GLn representation

Some motivation

Connecting branching with plethysm

Branching for symmetric powers

Using an existence result on the plethysm side