Cominuscule points and Schubert varieties

TL;DR

This paper introduces cominuscule points in Schubert varieties, providing formulas to compute Hilbert series and multiplicities using equivariant K-theory and cohomology, extending known results to broader cases.

Contribution

It generalizes formulas for Hilbert series and multiplicities at cominuscule points in Schubert varieties, applicable to more general schemes with torus actions.

Findings

Formulas for Hilbert series at cominuscule points

Formulas for multiplicities at cominuscule points

Extension to generalized cominuscule points

Abstract

We introduce the notion of a cominuscule point in a Schubert variety in a generalized flag variety for a semisimple group. We derive formulas expressing the Hilbert series and multiplicity of a Schubert variety at a cominuscule point in terms of the restrictions of classes in torus-equivariant K-theory and cohomology to that point, generalizing previously known formulas for flag varieties of cominuscule type. Thus, we can calculate Hilbert series and multiplicities in cases where these were previously unknown. The formulas for Schubert varieties are special cases of more general formulas valid at generalized cominuscule points of schemes with torus actions.