Duke Mathematical Journal

Puzzles and (equivariant) cohomology of Grassmannians

Allen Knutson and Terence Tao

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Abstract

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.

Article information

Source
Duke Math. J., Volume 119, Number 2 (2003), 221-260.

Dates
First available in Project Euclid: 23 April 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1082744732

Digital Object Identifier
doi:10.1215/S0012-7094-03-11922-5

Mathematical Reviews number (MathSciNet)
MR1997946

Zentralblatt MATH identifier
1064.14063

Subjects
Primary: 14N15: Classical problems, Schubert calculus
Secondary: 05E05: Symmetric functions and generalizations 05E10: Combinatorial aspects of representation theory [See also 20C30] 57R91: Equivariant algebraic topology of manifolds 57S25: Groups acting on specific manifolds

Citation

Knutson, Allen; Tao, Terence. Puzzles and (equivariant) cohomology of Grassmannians. Duke Math. J. 119 (2003), no. 2, 221--260. doi:10.1215/S0012-7094-03-11922-5. https://projecteuclid.org/euclid.dmj/1082744732


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