Title: Vertex models for the product of a Schur and Demazure polynomial
| Speaker: | Tim Miller |
| Affiliation: | University of Waterloo |
| Location: | MC 5479 |
There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm.
Abstract: Demazure atoms and characters are polynomials that each form a Z-basis for polynomials in n variables. The product of a Schur polynomial with a Demazure atom (resp. character) expands into a linear combination of Demazure atoms (resp. characters) with positive integer structure coefficients. There are known combinatorial rules that compute these coefficients using "skyline tableaux" given by Haglund, Luoto, Mason and Willigenburg. I have found alternative rules using the theory of integrable vertex models, inspired by a technique introduced by Zinn-Justin.
I use "coloured" vertex models for atoms and characters obtained from Borodin and Wheeler's models for non-symmetric Macdonald polynomials (setting q=t=0). The structure coefficients are then obtained as the number of fillings of a "diamond" vertex model that is compatible with both Schur (uncoloured) and Demazure (coloured) vertex models. The proof is completely combinatorial and very pretty.