Puzzles and (equivariant) cohomology of Grassmannians

Details Puzzles and (equivariant) cohomology of Grassmannians Puzzles and (equivariant) cohomology of Grassmannians

2001-12-14

arxiv.org/abs/math/0112150v1

Duke Math. J. 119 (2003), no. 2, 221--260

Mathematics

Algebraic Topology

30 pages, 21 figures

Scientific paper

We generalize our puzzle formula for ordinary Schubert calculus on Grassmannians, to a formula for the T-equivariant Schubert calculus. The structure constants to be calculated are polynomials in {y_{i+1} - y_i}; they were shown (abstractly) to have positive coefficients in [Graham] math.AG/9908172. Our formula is the first to be manifestly positive in this sense. In particular this gives a new and self-contained proof of the ordinary puzzle formula, by an induction backwards from the "most equivariant" case. The proof of the formula is mostly combinatorial, but requires no prior combinatorics, and only a modicum of equivariant cohomology (which we include). This formula is closely related to the one in [Molev-Sagan] q-alg/9707028 for multiplying factorial Schur functions in three sets of variables, although their rule does not give a positive formula in the sense of [Graham]. We include a cohomological interpretation of this problem, and a puzzle formulation for it.

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