Generalization of Cauchy's identity - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T07:28:19Z http://mathoverflow.net/feeds/question/114275 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114275/generalization-of-cauchys-identity Generalization of Cauchy's identity R. Rosenbaum 2012-11-23T18:34:40Z 2012-11-24T04:20:28Z <p>Let $s_{\lambda}$ be the schur function associated to the partition $\lambda$. Cauchy's identity (as in Macdonald) states that</p> <p>$$\sum_{\lambda} s_{\lambda}(X)s_{\lambda}(Y) = \prod_{i,j}(1-x_iy_j)^{-1}$$</p> <p>where the sum is over all partitions. There is a generalization appearing in a paper by Ishikawa and Tagawa <a href="http://www.uec.tottori-u.ac.jp/~mi/papers/fpsac07b.pdf" rel="nofollow">http://www.uec.tottori-u.ac.jp/~mi/papers/fpsac07b.pdf</a> (Theorem 2.1(i)) which states that if $X=(x_1,...,x_m)$ and $Y=(y_1,...,y_m)$</p> <p>$$\sum_{\lambda} w^{\lambda_m}s_{\lambda}(X)s_{\lambda}(Y) = \frac{1-|X||Y|}{(1-w|X||Y|)\prod_{i,j=1}^m(1-x_iy_j)}$$</p> <p>where the sum is over partitions $\lambda=(\lambda_1,...,\lambda_m)$ and $|X| = x_1x_2\cdots x_m$.</p> <p>I am curious if there is a result along these lines which gives a closed form product expression for the generating function</p> <p>$$\sum_{\lambda} z_1^{\lambda_1}\cdots z_{m-1}^{\lambda_{m-1}}z_m^{\lambda_m}s_{\lambda}(X)s_{\lambda}(Y)$$</p> <p>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</p> <p>$$\sum_{\lambda} w^{\lambda_{m-1}}z^{\lambda_m}s_{\lambda}(X)s_{\lambda}(Y)$$</p> <p>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.</p>