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

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.