Puzzles and (equivariant) cohomology of Grassmannians

The product of two Schubert cohomology classes on a Grassmannian Grk (C^n) has long been known to be a positive combination of other Schubert classes, and many manifestly positive formulae are now available for computing such a product (e.g., the LittlewoodRichardson rule or the more symmetric puzzle rule from A. Knutson, T. Tao, and C. Woodward [KTW]). Recently, W.Graham showed in [G], nonconstructively, that a similar positivity statement holds for Tequivariant} cohomology (where the coefficients are polynomials). We give the first manifestly positive formula for these coefficients in terms of puzzles using an ``equivariant puzzle piece.ʹʹThe proof of the formula is mostly combinatorial but requires no prior combinatorics and only a modicum of equivariant cohomology (which we include). As a by-product the argument gives a new proof of the puzzle (or Littlewood-Richardson) rule in the ordinary-cohomology case, but this proof requires the equivariant generalization in an essential way, as it inducts backwards from the "most equivariant" case.This formula is closely related to the one in A. Molev and B. Sagan [MS] for multiplying factorial Schur functions in three sets of variables, although their rule does not give a positive formula in the sense of [G]. We include a cohomological interpretation of their problem and a puzzle formulation for it.