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

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.