Combinatorics Seminar: K-theoretic Schubert Calculus for Orthogonal Orbits

Jiayi Wen, USC

Title: K-theoretic Schubert Calculus for Orthogonal Orbits

Abstract: Let G be a connected complex reductive group and B be a Borel subgroup. A subgroup H is called spherical if H acts on G/B with finitely many orbits. Brion gave an explicit formula to express the cohomology classes of H-orbits as linear combinations of Schubert cycles. The coefficients of Schubert cycles are nonnegative integers related to some combinatorial properties of weak orders. However, when one replaces the cohomology ring by the algebraic K-theory of coherent sheaves on G/B, the story becomes complicated and no general formula for H-orbits is available. In this talk, we will focus on the case when G is the general linear group, and H is the orthogonal subgroup in G. We will explore some new results in K-theory when the H-orbits are indexed by vexillary involutions, where the orbit closures have rational singularities.