Lowest sl(2)-types in sl(n)-representations with respect to a principal embedding

TL;DR

This paper proves that the smallest sl(2)-type dimension in any finite-dimensional sl(n)-representation with a principal embedding is exactly n, refining previous bounds and exploring other embeddings.

Contribution

It establishes that the minimal sl(2)-type dimension bound is exactly n for principal embeddings in sl(n), and investigates other embedding types.

Findings

The minimal sl(2)-type dimension bound is exactly n.

The bound b(n)=n is sharp for principal embeddings.

Methods involve Cartan--Helgason theorem, Pieri rules, and branching algebra calculations.

Abstract

Fix n>2. Let s be a principally embedded sl(2)-subalgebra in sl(n). A special case of results of the second author and Gregg Zuckerman implies that there exists a positive integer b(n) such that for any finite dimensional sl(n)-representation, V, there exists an irreducible s-representation embedding in V with dimension at most b(n). We prove that b(n)=n is the sharpest possible bound. We also address embeddings other than the principal one. The exposition involves an application of the Cartan--Helgason theorem, Pieri rules, Hermite reciprocity, and a calculation in the "branching algebra" introduced by Roger Howe, Eng-Chye Tan, and the second author.