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.

"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).