Positivity of Equivariant Schubert Classes through Moment Map Degeneration
نویسنده
چکیده
For a flag manifold M = G/B with the canonical torus action, the T−equivariant cohomology is generated by equivariant Schubert classes, with one class τu for every element u of the Weyl group W . These classes are determined by their restrictions to the fixed point set M ≃ W , and the restrictions are polynomials with nonnegative integer coefficients in the simple roots. The main result of this article is a positive formula for computing τu(v) in types A, B, and C. To obtain this formula we identify G/B with a generic co-adjoint orbit and use a result of Goldin and Tolman to compute τu(v) in terms of the induced moment map. Our formula, given as a sum of contributions of certain maximal ascending chains from u to v, follows from a systematic degeneration of the moment map, corresponding to degenerating the co-adjoint orbit. In type A we prove that our formula is manifestly equivalent to the formula announced by Billey in [Bi], but in type C, the two formulas are not equivalent.
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