| Speaker: | Anton Khoroshkin (University of Haifa, Israel) |
|---|---|
| Title: | Cauchy Identities and Howe Duality for Staircase Matrices |
| Date (JST): | Thu, Sep 24, 2026, 13:30 - 15:00 |
| Place: | Seminar Room B |
| Abstract: |
The classic Cauchy identity allows us to write a product of terms $(1 - x_i y_j)^{-1}$ over the entries of a rectangular matrix as a sum of products of Schur polynomials in $x$ and $y$. This famous formula is a direct result of Howe duality, which describes the decomposition into irreducibles of the polynomial ring of rectangular matrices as a $(\mathfrak{gl}_m, \mathfrak{gl}_n)$-bimodule. In this talk, we move beyond rectangles to explore staircase-shaped matrices. By studying how upper-triangular (Borel) matrices act on these shapes, we find a new, more general version of the Cauchy identity. I will explain this generalized Howe duality and show how the terms in our new sum reveal a rich and surprising connection to the (parabolic) Bruhat graph of the symmetric group and the Bubble sort algorithm. The talk is based on joint projects with Ie. Makedonskyi and E.Feigin (details can be found in arxiv:2502.21184 and in arxiv:2411.03117 ). |
