The product of two Schubert cohomology classes on a Grassmannian
${\rm Gr}_k (\mathbb{c}^n)$ 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
Littlewood-Richardson 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 {\em $T$-equivariant} 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.''
¶ The 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.