$\hbar$-deformed Schubert calculus in equivariant cohomology, K-theory, and elliptic cohomology

TL;DR

This survey reviews recent progress in $$-deformed Schubert calculus across equivariant cohomology, K-theory, and elliptic cohomology, highlighting formulas, relations, and explicit structure constants.

Contribution

It synthesizes recent developments in $$-deformed classes of Schubert varieties, connecting them through stable envelopes and providing explicit formulas for structure constants.

Findings

Derived explicit formulas for $$-deformed classes.

Established orthogonality relations for these classes.

Connected different $$-deformed theories via stable envelopes.

Abstract

In this survey paper we review recent advances in the calculus of Chern-Schwartz-MacPherson, motivic Chern, and elliptic classes of classical Schubert varieties. These three theories are one-parameter ($\hbar$) deformations of the notion of fundamental class in their respective extraordinary cohomology theories. Examining these three classes in conjunction is justified by their relation to Okounkov's stable envelope notion. We review formulas for the $\hbar$-deformed classes originating from Tarasov-Varchenko weight functions, as well as their orthogonality relations. As a consequence, explicit formulas are obtained for the Littlewood-Richardson type structure constants.