Equivariant rigidity of Richardson varieties

TL;DR

This paper demonstrates that Schubert and Richardson varieties are uniquely identified by their equivariant cohomology, and explores implications for quantum cohomology and K-theory, including a conjecture proven for certain flag varieties.

Contribution

It establishes the equivariant rigidity of Richardson varieties and introduces a conjecture linking Seidel classes to equivariant quantum K-theory, proven for cominuscule flag varieties.

Findings

Schubert and Richardson varieties are uniquely determined by their equivariant cohomology classes.

A stronger result relates Bialynicki-Birula cell closures to Richardson varieties.

The conjecture on Seidel multiplication in equivariant quantum K-theory is proven for cominuscule flag varieties.

Abstract

We prove that Schubert and Richardson varieties in flag manifolds are uniquely determined by their equivariant cohomology classes, as well as a stronger result that replaces Schubert varieties with closures of Bialynicki-Birula cells under suitable conditions. This is used to prove that any two-pointed curve neighborhood representing a quantum cohomology product with a Seidel class is a Schubert variety. We pose a stronger conjecture which implies a Seidel multiplication formula in equivariant quantum K-theory, and prove this conjecture for cominuscule flag varieties.