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15 August 2003 Puzzles and (equivariant) cohomology of Grassmannians
Allen Knutson, Terence Tao
Duke Math. J. 119(2): 221-260 (15 August 2003). DOI: 10.1215/S0012-7094-03-11922-5


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.


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Allen Knutson. Terence Tao. "Puzzles and (equivariant) cohomology of Grassmannians." Duke Math. J. 119 (2) 221 - 260, 15 August 2003.


Published: 15 August 2003
First available in Project Euclid: 23 April 2004

MathSciNet: MR1997946
zbMATH: 1064.14063
Digital Object Identifier: 10.1215/S0012-7094-03-11922-5

Primary: 14N15
Secondary: 05E05 , 05E10 , 57R91 , 57S25

Rights: Copyright © 2003 Duke University Press


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Vol.119 • No. 2 • 15 August 2003
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