I apologize if this question is trivial, but a couple of days of searching for necessary routines have led me here.

Does there exist software to compute symmetric powers of Schur polynomials?

I am seeking such software in the hopes of computing the characters of representations of the simple Lie algebra $A_n$, i.e., $S_\lambda(x_1, \dots, x_n)$ to use the notation of Fulton and Harris, and then applying $\mathrm{Sym}^k$ to the resulting Schur polynomial and writing the result as a sum of Schur polynomials corresponding to differing partitions $\mu$. That is, as an example, I would like to compute something like the following: $$ \mathrm{Sym}^3(S_\lambda(x_1, \dots, x_4)) = \sum_\mu k_\mu S_\mu(x_1, \dots, x_4) $$ Where $S_\lambda(x_1, \dots, x_4)$ denotes the character of the irreducible representation of $A_4$ with highest weight $\lambda$. I understand that Mathematica may compute symmetric polynomials, however I have not found any routines for applying $\mathrm{Sym}$ to these polynomials. Regards.