Irreducible representations of the symmetric group Sym$(n)$, and degree-$n$ algebraic representations of SL$\_d(\mathbb C)$ for $d\ge n$, can both be classified by Young diagrams with $n$ boxes.


Consider now a degree-$n$ irreducible representation of SL$\_d(\mathbb C)$ with $d\ge n$, and restrict it to the subgroup of permutation matrices Sym$(d)$. What is its decomposition in irreducibles? Is there a nice way of listing those Young diagrams (with $d$ boxes) which occur, constructed out of the Young diagram with $n$ boxes of the original representation?



Representations of Sym(n) and SL_d
This is given in Exercise 7.74 of Enumerative Combinatorics II by Richard Stanley which is a formula for the restrictions in terms of plethysm and inner product.


