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

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