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 $n?$
