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.



Symmetric powers of Schur polynomials
This can be done with LiE:
<http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/>


(In fact it will compute the Schur functor of any irreducible representation.) There is a form interface so you can try LiE on the web: <http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/form.html>


Here is an example of calculating $Sym^3$ of $s\_{2,1}$ (everything is written in fundamental weight notation, so X[1,1,0] below refers to the partition (2,1,0) = (1,0,0) + (1,1,0)):


Input: sym\_tensor(3,X[1,1,0],A3)


Output: 1X[0,0,3] +1X[0,1,1] +1X[0,3,1] +1X[1,0,0] +1X[1,1,2] +1X[1,2,0] + 2X[2,0,1] +1X[2,2,1] +1X[3,0,2] +1X[3,1,0] +1X[3,3,0]


The only caveat is that LiE treats A3 as $SL\_4$, so for instance, the partition (2,1,1,1) is the same as the partition (1,0,0,0) (because we have the identification $x\_1x\_2x\_3x\_4 = 1$).


