Chern class formulas for $G_{2}$ Schubert loci

Abstract

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 BernsteinGelfand-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, correcting an error in a paper by Edidin and Graham.

Quantum Schubert polynomials for the G2 flag manifold

“…In §5 we use the Chevalley rule to find Schubert polynomials for σ w . Some of our polynomials coincide with similar Schubert polynomials found by D. Anderson [1], via different methods. The polynomials we found are homogeneous and have positive coefficients.…”

“…In order to calculate P s2s1 we use the identity σ s1 σ s1 = σ s2s1 + q 1 (taken from Table 4). Using that Ψ is an algebra homomorphism, we know that Ψ(x Computations of ordinary Schubert polynomials were done for the ordinary cohomology ring H * (X) of the G 2 flag manifold in a paper by Anderson [1]. A classical result of Borel [2] shows that H Table 5.…”

Degeneracy of triality-symmetric morphisms

“…Let −x 1 , −x 2 be Chern roots of E, so x 1 , x 2 are Chern roots of E * . Then, by [Anderson 2011, Theorem 2.4 and Section 2.5], we have…”

“…Flag bundles and Schubert loci. We refer to [Anderson 2009;2011] for proofs of the following facts with more details. (There the term "γ -isotropic" is used instead of "G 2 -isotropic" in reference to a trilinear form γ .…”

“…The first relates degeneracy loci for triality-symmetric morphisms to certain Schubert loci in a G 2 flag bundle, just as Fulton's generalization of the Harris-Tu formulas relates symmetric morphisms to type C flag bundles [Fulton 1996]. One then applies the formulas for G 2 Schubert loci developed in [Anderson 2011] to derive the formulas of Theorem 1.2.…”