Degeneracy Loci Classes in $K$-theory - Determinantal and Pfaffian Formula -

TL;DR

This paper establishes determinantal and Pfaffian formulas for $K$-theoretic degeneracy loci classes in various bundles, extending classical formulas and introducing new factorial functions for equivariant $K$-theory.

Contribution

It generalizes classical determinantal and Pfaffian formulas to $K$-theory and introduces factorial $G heta$-functions for equivariant $K$-theoretic Schubert classes.

Findings

Derived determinantal and Pfaffian formulas for degeneracy loci classes.

Introduced factorial $G heta$-functions for torus equivariant $K$-theory.

Extended classical formulas to the $K$-theoretic setting.

Abstract

We prove a determinantal formula and Pfaffian formulas that respectively describe the $K$-theoretic degeneracy loci classes for Grassmann bundles and for symplectic Grassmann and odd orthogonal bundles. The former generalizes Damon--Kempf--Laksov's determinantal formula and the latter generalize Pragacz--Kazarian's formula for the Chow ring. As an application, we introduce the factorial $G\Theta / G\Theta'$-functions representing the torus equivariant $K$-theoretic Schubert classes of the symplectic and the odd orthogonal Grassmannians, which generalize the (double) theta polynomials of Buch--Kresch--Tamvakis and Tamvakis--Wilson.