Lowest 𝔰𝔩(2)-types in 𝔰𝔩(𝔫)-representations

Abstract

Fix n ≥ 3 n \geq 3 . Let s \mathfrak {s} be a principally embedded s l 2 \mathfrak {sl}_2 -subalgebra in s l n \mathfrak {sl}_n . A special case of results of the second author and Gregg Zuckerman implies that there exists a positive integer b ( n ) b(n) such that for any finite dimensional irreducible s l n \mathfrak {sl}_n -representation, V V , there exists an irreducible s \mathfrak {s} -representation embedding in V V with dimension at most b ( n ) b(n) . We prove that b ( n ) = n 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.