  Let S be a principally embedded sl_2 subalgebra in sl_n for n > 2. A special
case of results of the third 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). In a 2017 paper (joint with Hassan Lhou),
they prove that b(n)=n is the sharpest possible bound, and also address
embeddings other than the principal one.
  These results concerning embeddings may by interpreted as statements about
plethysm. Then, a well known result about these plethysms can be interpreted as
a "branching rule". Specifically, a (finite dimensional) representation of
GL(n,C) will decompose into irreducible representations of the symmetric group
when it is restricted to the subgroup consisting of permutation matrices. The
question of which irreducible representations of the symmetric group occur with
positive multiplicity is the topic of this paper, applying the previous work of
Lhou, Zuckerman, and the third author.
