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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.

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  • $\begingroup$ How is $Sym^k$ defined? How would you do this by hand on a small example? $\endgroup$ May 29 '13 at 19:52
  • $\begingroup$ Let's consider the representation of $A_2$ i.e., $\mathfrak{sl}_3$ given by $\lambda = (3,2)$ so this has highest weight $\omega_1 + 2\omega_2$ where $\omega_i$ is the $i$th fundamental weight of $A_2$, and I write an arbitrary irreducible representation of $A_2$ as $\Gamma_{a,b}$ where $\lambda = a\omega_1 + b\omega_2$ is the highest weight. Then $Sym^2$ applied to this Schur polynomial $s_{(3,2)}$ should yield the Schur polynomials $s_{(2,2)} + s_{(3,1)} + s_{(4,3)} + s_{(4,0)} + s_{(6,4)}$. Is this what you meant? $\endgroup$
    – Moderat
    May 29 '13 at 21:05
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    $\begingroup$ The right keyword is "plethysm" $\endgroup$ May 29 '13 at 21:10
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    $\begingroup$ You can use John Stembridge's SF package for Maple: dept.math.lsa.umich.edu/~jrs/maple.html $\endgroup$
    – Ira Gessel
    May 29 '13 at 22:40
  • $\begingroup$ Josh Izzard: I have worked a bit with Schur polys from the combinatorial viewpoint, but my representation theory is almost non-existent. The word "plethysm" rings a bell... What is the rule for getting (2,2),(3,1) and so on from (3,2)? Or is this what you are looking for? Can these be described as "All partitions such that ...blah blah blah, in relation to (3,2) is true"? $\endgroup$ May 30 '13 at 22:15
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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 $x_1x_2x_3x_4 = 1$).

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  • $\begingroup$ Many thanks Steven. I will look into this interface $\endgroup$
    – Moderat
    Jun 7 '13 at 2:04
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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)
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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 $x_1,x_2,x_3,x_4$.

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