| We define degeneracy loci for vector bundles with structure group G2, 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 projective 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 G2. We include explicit descriptions of the G2 flag variety and its Schubert varieties, and several computations, including one that answers a question of William Graham.;As part of our description of the G2 flag variety, we prove some basic facts about octonions and trilinear forms, and give a natural construction of octonion algebra bundles which appears to be new.;Motivated by the relationship between symmetric matrices and the symplectic group, we define a new type of symmetry for morphisms of vector bundles, called triality symmetry. We explain the relation with G2, and deduce degeneracy locus formulas for triality-symmetric morphisms from formulas for Schubert loci in G2 flag bundles. We also give a proof of the formulas in terms of equivariant cohomology, by computing the classes of P-orbits in g2/p for a parabolic subgroup P ⊂ G 2.;In five appendices, we collect some facts from representation theory; review the phenomenon of triality and its relation to G 2 flags; discuss a general notion of symmetry for morphisms of vector bundles; give parametrizations of Schubert cells, formulas for degeneracy loci, and the equivariant multiplication table for the G 2 flag variety; and compute the Chow rings of quadric bundles. |
