Explicit Formulas for h-Deformed Structure Constants of Grassmannians

TL;DR

This paper develops explicit formulas for the structure constants of deformed classes in Grassmannians, using combinatorics and residue calculus, and proposes conjectures for broader cases.

Contribution

It introduces explicit formulas for structure constants of deformed classes in Grassmannians and related spaces, advancing the understanding of their algebraic structure.

Findings

Formulas for structure constants in projective spaces involving binomial coefficients

Residue calculus approach to structure constants of projective spaces and 2-plane Grassmannians

Conjectures for generalization to broader classes of Grassmannians

Abstract

The Chern-Schwartz-MacPherson (CSM) and motivic Chern (mC) classes of Schubert cells in a Grassmannian are one parameter deformations of the fundamental classes of the Schubert varieties in cohomology and K-theory respectively. Like the fundamental classes, the deformed classes form a basis for the cohomology and K-theory ring of the Grassmannian. The purpose of this paper is to initiate the study of the structure constants associated to the basis CSM and mC classes in terms of the combinatorics of polynomials. First, we prove formulas for the structure constants of projective spaces that involve binomial coefficients. Then, using residue calculus on wieght functions, we describe the structure constants of projective spaces and certain related structure constants of 2-plane Grassmannians as coefficients of explicit polynomials in one variable. Finally, we propose an approach for obtaining more general results in this direction, and make conjectures generalizing the aformentioned results for projective spaces.