# Generalization of Cauchy's identity

Let $s_{\lambda}$ be the schur function associated to the partition $\lambda$. Cauchy's identity (as in Macdonald) states that

$$\sum_{\lambda} s_{\lambda}(X)s_{\lambda}(Y) = \prod_{i,j}(1-x_iy_j)^{-1}$$

where the sum is over all partitions. There is a generalization appearing in a paper by Ishikawa and Tagawa http://www.uec.tottori-u.ac.jp/~mi/papers/fpsac07b.pdf (Theorem 2.1(i)) which states that if $X=(x_1,...,x_m)$ and $Y=(y_1,...,y_m)$

$$\sum_{\lambda} w^{\lambda_m}s_{\lambda}(X)s_{\lambda}(Y) = \frac{1-|X||Y|}{(1-w|X||Y|)\prod_{i,j=1}^m(1-x_iy_j)}$$

where the sum is over partitions $\lambda=(\lambda_1,...,\lambda_m)$ and $|X| = x_1x_2\cdots x_m$.

I am curious if there is a result along these lines which gives a closed form product expression for the generating function

$$\sum_{\lambda} z_1^{\lambda_1}\cdots z_{m-1}^{\lambda_{m-1}}z_m^{\lambda_m}s_{\lambda}(X)s_{\lambda}(Y)$$

where the sum is still over partitions $\lambda=(\lambda_1,...,\lambda_m)$. In particular, at least for now I care about such a generating function in the form

$$\sum_{\lambda} w^{\lambda_{m-1}}z^{\lambda_m}s_{\lambda}(X)s_{\lambda}(Y)$$

It is my understanding that such a result can likely be discovered via the RSK correspondence, but this is not really my field. Thus before I delve into possibly slowly reinventing the wheel I thought I'd ask.

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