Symmetric powers of Schur polynomials 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.
 A: this could be done in sage:
sage: B3 = WeylCharacterRing("B3", style="coroots")
sage: spin = B3(0,0,1)
sage: spin.symmetric_power(6)
B3(0,0,0) + B3(0,0,2) + B3(0,0,4) + B3(0,0,6)

sage: A3 = WeylCharacterRing("A3", style="coroots")
sage: rep = A3(0,0,1)              
sage: rep.symmetric_power(6)     
A3(0,0,6)

A: 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 $x_1x_2x_3x_4 = 1$).
A: If I am not mistaken, you are interested in the plethysm
$$
 h_k[ s_\lambda]
$$
If you prefer Mathematica, I have a package for this type of computations. For example, $Sym^3(S_{21})$ is computed as
ToSchurBasis@Plethysm[CompleteHSymbol[3],SchurSymbol[{2,1}]]

gives
$$
s_{63}+s_{333}+s_{432}+s_{441}
   +s_{522}+s_{531}+s_{3222}+s
   _{3321}+2
   s_{4221}+s_{4311}+s_{5211}+
   s_{32211}+s_{33111}+s_{4211
   1}+s_{411111}
$$
which matches Steven Sam's result (I believe).
Note that the output consists of Schur symmetric functions and not polynomials, so all the Schur functions indexed by partitions with length more than 4 are zero if you only work in $x_1,x_2,x_3,x_4$.
