Symmetric powers of Schur polynomials

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)

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 $x_1x_2x_3x_4 = 1$).