There has been considerable recent work to extend the algebra and geometry of Schubert polynomials from the type A case originated by Lascoux and Schutzenberger to types B, C, and D. Single polynomials were given by Billey and Haiman, and double polynomials by Ikeda, Mihalcea, and Naruse. Although these give many degeneracy formulas, they do not suffice even for the classical case of ranks of symmetric or skew-symmetric matrices of homogeneous polynomials on projective spaces.

For these one needs to include symplectic and quadratic forms with values in a line bundle. In joint work with Dave Anderson, we have constructed "twisted double Schubert polynomials" which have a new parameter corresponding to the line bundle. They are in twisted versions of the classical rings, which correspond to new and stable presentations of cohomology/Chow rings of isotropic Grassmann bundles in these types. We show that that whole package of Type A phenomena, including positivity assertions, extend to these polynomials. Speaker(s): William Fulton (University of Michigan)