Degeneracy loci and G2 Flags.

TL;DRAbstract

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 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 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 William Graham. As part of our description of the G_2 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 def

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