The classic Cauchy identity expresses the product $\prod_{i,j} (1 - x_iy_j)^{-1}$ over the entries of a rectangular matrix as a sum of products of Schur polynomials in $x$ and $y$. This fundamental formula arises directly from Howe duality, which governs the decomposition of the polynomial ring of rectangular matrices into irreducibles as a $(\mathfrak{gl}_m, \mathfrak{gl}_n)$-bimodule.

In this talk, we move beyond rectangular matrices to explore staircase-shaped matrices. By studying the action of upper-triangular (Borel) matrices on these shapes, we establish a novel generalization of the Cauchy identity. I will explain this generalized Howe duality and demonstrate how the terms in the resulting sum reveal a rich connection to the (parabolic) Bruhat graph of the symmetric group and the bubble sort algorithm.

This talk is based on joint works with I. Makedonskyi and E. Feigin (arXiv:2502.21184, arXiv:2411.03117).

Place: Room 003, Department of Botany Annexe, RIMS