Branching from the General Linear Group to the Symmetric Group and the Principal Embedding

Let S be a principally embedded ???????? 2 -subalgebra in n for n?3. A special case of results the third author and Gregg Zuckerman implies that there exists positive integer b(n) such any finite-dimensional irreducible -representation, V, an S-representation embedding V with dimension at most b(n). In 2017 paper (joint Hassan Lhou), they prove b(n)=n is sharpest possible bound, also address embeddings other than principal one.