表現論セミナー Colored Vertex Models and Grothendieck Polynomials（Travis Scrimshaw）

開催日時

2022年7月14日 16時30分 ～ 2022年7月14日 18時00分

場所

理学部3-413

講演者

Travis Scrimshaw氏 (北大)

Schubert calculus is a classical subject is algebraic geometry related to a decomposition of the Grassmannian (k-dimensional planes in n-dimensional space) and the flag variety (maximal length sequences of increasing subspaces of n-dimensional space) into orbits of the upper triangular matrices. These give rise to cohomology classes that can be represented by Schur and Schubert polynomials (up to an ideal), respectively. Both of these have been extremely well-studied with multiple viewpoints. If instead we look at the K-theory ring, we obtain representatives given by (symmetric) Grothendieck polynomials, which are no longer homogeneous polynomials and are less studied. However, they can be defined by divided difference operators like for the Schur and Schubert polynomials. In this talk, we will construct colored vertex models whose partition functions are (up to a simple overall factor) the (symmetric) Grothendieck polynomials, which have a natural connection to the underlying combinatorics. No knowledge will be assumed. This is based upon joint work with Valentin Buciumas and Katherine Weber (arXiv:2007.04533, J. Lond. Math. Soc.; arXiv:2007.04533, IMRN).