Let $V$ be the irreducible representation of $GL\_n$ with highest $\lambda$, and $|\lambda|=n$. It is well known that the representation of $S\_n$ on the $(1,1,\ldots,1)$ weight space is the Specht module $S\_{\lambda}$. What is known about the representation of $S\_n$ on the rest of $V$?


I am also interested in the case where $|\lambda| < n$.



Restriction from $GL_n$ to $S_n$
I keep meaning to post this and forgetting: Richard Stanley sent me the following two references


*Enumerative Combinatorics, Volume II* Exercise 7.74: If $V\_{\lambda}$ is a representation of $GL\_n$, and $S\_{\mu}$ a representation of $S\_n$, then the multiplicity of $S\_{\mu}$ in $\mathrm{Restriction}^{GL\_n}\_{S\_n} V\_{\lambda}$ is the coefficient of the Schur function $s\_{\lambda}$ in the symmetric formal power series
$$s\_{\mu}(1, x\_1, x\_2, \ldots, x\_1^2, x\_1 x\_2, x\_1 x\_3, \ldots, x\_1^3, x\_1^2 x\_2, \ldots).$$
That is to say, we take the Schur function $s\_{\mu}$ and plug in every monomial. (Yes, for those who know the term, this is an example of plethysm.)


[Gay, David A.](http://www.ams.org/mathscinet-getitem?mr=414794)   

"Characters of the Weyl group of $SU(n)$ on zero weight spaces and centralizers of permutation representations."   

*Rocky Mountain J. Math.* **6** (1976), no. 3, 449—455. 


Determines the representation of $S\_n$ on the zero weight space of $V\_{\lambda}$ when $|\lambda|$ is a multiple of $n$.


I just found another reference, which I haven't digested yet: [Scharf, Thibon and Wybourne (1997)](http://iopscience.iop.org/0305-4470/30/19/030/).


