Consider the generating function $$ G_n(x_1,x_2,\ldots,x_n, t_1,t_2,\ldots,t_n) =\sum_{\lambda}s_{\lambda}(x_1,x_2,\ldots, x_n) t_1^{\lambda_1}t_2^{\lambda_2} \cdots t_n^{\lambda_n}, $$ where the sum is over all partitions $\lambda=(\lambda_1, \lambda_2,\ldots, \lambda_n)$ and $s_\lambda$ is a Schur polynomial.

For small $n$ such generating function is easy to find, for example for $n=2$ by direct calculation we have $$ G_2(x_1,x_2, t_1,t_2)=\sum_{\lambda}s_{\lambda}(x_1,x_2) t_1^{\lambda_1}t_2^{\lambda_2}=\frac{1}{(1-x_1 t_1)(1-x_2 t_1)(1-x_1 x_2 t_1 t_2)}. $$ If we put $t_1=t_2=1$ then we come to well-known Littlewood identity

$$ \sum_{\lambda}s_{\lambda}(x_1,x_2)=\frac{1}{(1-x_1)(1-x_2)(1-x_1 x_2)}. $$

Question. Is there any close expression for the generating function $G_n$ for arbitrary $n?$


1 Answer 1


This is done in my paper The character generator of SU(n). I believe there was an essentially the same previous MO question, but I am unable to find it.

  • $\begingroup$ In case the link to the PDF file is broken, the DOI is (almost certainly) doi.org/10.1063/1.524687 $\endgroup$ Dec 20, 2021 at 13:46
  • $\begingroup$ @Richard Stanley. Thank you for the answer! $\endgroup$
    – Leox
    Dec 20, 2021 at 17:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.