Residue formula for flag manifold of type $A$ from wall-crossing

TL;DR

This paper re-proves residue formulas for equivariant integrals on type A flag manifolds using wall-crossing inspired methods and applies these results to derive determinantal formulas for Grothendieck polynomials.

Contribution

It introduces a wall-crossing inspired computational approach to re-derive residue formulas and applies it to K-theory classes for Grothendieck polynomials.

Findings

Re-proved residue formulas for equivariant integrals on flag manifolds.

Derived determinantal formulas for Grothendieck polynomials.

Connected wall-crossing techniques with algebraic geometry computations.

Abstract

We consider equivariant integrals on flag manifolds of type $A$. Using a computational method inspired by the theory of wall-crossing formulas by Takuro Mochizuki, we re-prove residue formulas for equivariant integrals given by Weber and Zielenkiewicz. As an application, we give the determinantal formula of the Grothendieck polynomial by properly setting $K$ theory classes.