K-theoretic Chern class formulas for vexillary degeneracy loci

TL;DR

This paper develops K-theoretic formulas for degeneracy loci in classical types using raising operators, providing new determinantal and Pfaffian expressions for rank conditions, and introduces a refined Euler class for quadratic bundles.

Contribution

It introduces novel K-theoretic formulas for degeneracy loci and new determinantal and Pfaffian expressions for classical rank conditions, extending previous results.

Findings

Derived K-theoretic formulas for classical degeneracy loci.

Established new determinantal and Pfaffian expressions for rank conditions.

Constructed a K-theoretic Euler class for quadratic bundles.

Abstract

Using raising operators and geometric arguments, we establish formulas for the K-theory classes of degeneracy loci in classical types. We also find new determinantal and Pfaffian expressions for classical cases considered by Giambelli: the loci where a generic matrix drops rank, and where a generic symmetric or skew-symmetric matrix drops rank. In an appendix, we construct a K-theoretic Euler class for even-rank vector bundles with quadratic form, refining the Chow-theoretic class introduced by Edidin and Graham. We also establish a relation between top Chern classes of maximal isotropic subbundles, which is used in proving the type D degeneracy locus formulas.