# Generating function for Schur polynomials

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 $$n?$$

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.

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