Cauchy identity for staircase matrices and related combinatorics

The classical Cauchy identity expresses the product $(1 - x_i y_j)^{-1}$, where the indices (i,j) run over the entries of a rectangular matrix, as a sum over partitions (\lambda) of products of Schur polynomials $s_{\lambda}(x) s_{\lambda}(y)$. By breaking the symmetry, we describe a decomposition of the product $(1 - x_i y_j)^{-1}$, where (i,j) ranges over the cells of a staircase matrix, in terms of a sum of products of key polynomials, $\kappa_{\mu}(x) \kappa_{\nu}(y)$, for certain pairs of compositions (\mu,\nu). The pairs (\mu,\nu) appearing in the decomposition are determined combinatorially by rook placements in the staircase matrix, and by the bubble-sort algorithm.

The aim of this talk is to explain all the combinatorial ingredients behind this “generalized Cauchy identity,” which I hope will be of independent interest to participants of the seminar and does not require any background from the theory of symmetric functions.

The talk is based on joint works with Ie. Makedonskyi arXiv:2502.21184 and E. Feigin arXiv:2411.03117.