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

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 $G\Theta /G{\Theta }^{\prime }$-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.

This is a major update. In particular, we extended the results in arXiv:1602.04448 and included in this version. We modified the expositions in several places from the previous version

References in corpus (11)

• Algebraic cobordism
• Chern class formulas for classical-type degeneracy loci
• Degeneracy Loci, Pfaffians, and Vexillary Signed Permutations in Types B, C, and D
• Double eta polynomials and equivariant Giambelli formulas
• Double theta polynomials and equivariant Giambelli formulas
• Construction of double Grothendieck polynomials of classical types using IdCoxeter algebras
• Interpolation analogues of Schur Q-functions
• Segre classes and Kempf-Laksov formula in algebraic cobordism
• Connective algebraic K-theory
• Pfaffian sum formula for the symplectic Grassmannians
• Generalised symplectic Schubert classes

Cited by in corpus (20)

• Equivariant K-theory and refined Vafa-Witten invariants
• Double theta polynomials and equivariant Giambelli formulas
• K-theory formulas for orthogonal and symplectic orbit closures
• Double eta polynomials and equivariant Giambelli formulas
• A symplectic refinement of shifted Hecke insertion
• Grothendieck polynomials and the Boson-Fermion correspondence
• K-theoretic crystals for set-valued tableaux of rectangular shapes
• Free-fermions and skew stable Grothendieck polynomials
• K-classes of Brill-Noether loci and a determinantal formula
• Elementary proof and application of the generating function for generalized Hall-Littlewood functions
• Segre classes and Kempf-Laksov formula in algebraic cobordism
• Double Grothendieck Polynomials for Symplectic and Odd Orthogonal Grassmannians
• Algebraic Spivak's theorem and applications
• Neutral-fermionic presentation of the $K$-theoretic $Q$-function
• Back stable K-theory Schubert calculus
• Kempf-Laksov Schubert classes for even infinitesimal cohomology theories
• Darondeau-Pragacz formulas in complex cobordism
• Pfaffian formula for $K$-theory of odd orthogonal Grassmannians
• Integrable models and $K$-theoretic pushforward of Grothendieck classes
• Symplectic and odd orthogonal Pfaffian formulas for algebraic cobordism