The symmetric group $S_n$ on n letters is embedded in the general linear group $GL_n(C)$ as a subgroup of permutation matrices. When an irreducible representation (Weyl module) of $GL_n(C)$ is restricted to $S_n$, it decomposes into a direct sum of irreducible representations of $S_n$, with certain irreducible components potentially appearing multiple times. The restriction problem, an open problem in algebraic combinatorics, seeks a combinatorial interpretation of these multiplicities, known as restriction coefficients.
On the other hand, polynomials in an infinite sequence of variables can be evaluated as class functions of $S_n$ for all n. When these polynomials represent characters of families of representations, they are referred to as character polynomials.
In a joint work with Narayanan, Prasad, and Srivastava, we compute character polynomials and their moments for Weyl modules. Using these results, we interpret certain restriction coefficients as signed sums of vector partitions. Finally, we show how moment-generating functions can be applied to solve the restriction problem for Weyl modules indexed by hook partitions.
