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 $n?$



Generating function for Schur polynomials
This is done in my paper [The character generator of SU(*n*)](https://klein.mit.edu/~rstan/pubs/pubfiles/44.pdf). I believe there was an essentially the same previous MO question, but I am unable to find it.


