Abstract. Fix n ≥ 3. 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 irreducible 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.
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