Chern class formulas for $G_2$ Schubert loci

TL;DR

This paper develops formulas for the cohomology classes of degeneracy loci associated with $G_2$-structured vector bundles, extending classical Schubert calculus to the exceptional Lie group $G_2$ with explicit computations.

Contribution

It introduces the first explicit Chern class formulas for $G_2$ Schubert loci, extending classical degeneracy locus formulas to the exceptional group case.

Findings

Formulas for $G_2$ degeneracy loci classes in cohomology and Chow groups.

Explicit descriptions of $G_2$ flag varieties and Schubert varieties.

Computations answering a question of W. Graham and clarifying Chow ring structures.

Abstract

We define degeneracy loci for vector bundles with structure group $G_2$, and give formulas for their cohomology (or Chow) classes in terms of the Chern classes of the bundles involved. When the base is a point, such formulas are part of the theory for rational homogeneous spaces developed by Bernstein-Gelfand-Gelfand and Demazure. This has been extended to the setting of general algebraic geometry by Giambelli-Thom-Porteous, Kempf-Laksov, and Fulton in classical types; the present work carries out the analogous program in type $G_2$. We include explicit descriptions of the $G_2$ flag variety and its Schubert varieties, and several computations, including one that answers a question of W. Graham. In appendices, we collect some facts from representation theory and compute the Chow rings of quadric bundles, clarifying a previous computation of Edidin and Graham.