2001/12/14 by Allen Knutson, Terence Tao · 2 citations
Mathematics · #math.AT #math.AG #math.CO #msc:14N15 #msc:57S25 #msc:57R91
published as Duke Math. J. 119 (2003), no. 2, 221--260 · 30 pages, 21 figures
arxiv created 2001/12/14 · arxiv updated 2010/04/26
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 yi+1 - yi; 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.