Duke Mathematical Journal

Puzzles and (equivariant) cohomology of Grassmannians

Allen Knutson and Terence Tao

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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

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

First available in Project Euclid: 23 April 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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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  • A. Arabia, Cohomologie $T$-équivariante de $G/B$ pour un groupe $G$ de Kac-Moody, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), 631--634.
  • A. Bia łynicki-Birula, Some properties of the decompositions of algebraic varieties determined by actions of a torus, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), 667--674.
  • A. Buch, The saturation conjecture (after A. Knutson and T. Tao), with an appendix by William Fulton, Enseign. Math. (2) 46 (2000), 43--60.
  • --. --. --. --., A Littlewood-Richardson rule for the K-theory of Grassmannians, Acta Math. 189 (2002), 37--78. \CMP1 946 917
  • M. Demazure, ``Désingularisation des variétés de Schubert généralisées'' in Collection of Articles Dedicated to Henri Cartan on the Occasion of His 70th Birthday, Ann. Sci. École Norm. Sup. (4) 7, Gauthier-Villars, Paris, 1974, 53--88.
  • W. Fulton, Young Tableaux: With Applications to Representation Theory and Geometry, London Math. Soc. Student Texts 35, Cambridge Univ. Press, Cambridge, 1997.
  • M. Goresky, R. Kottwitz, and R. MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), 25--83.
  • W. Graham, Positivity in equivariant Schubert calculus, Duke Math. J. 109 (2001), 599--614.
  • V. Guillemin and C. Zara, ``Equivariant de Rham theory and graphs'' in Sir Michael Atiyah: A Great Mathematician of the Twentieth Century, Asian J. Math. 3, Internat. Press, Somerville, Mass., 1999, 49--76.
  • A. Knutson and T. Tao, The honeycomb model of $\GL(n)$ tensor products, I: Proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999), 1055--1090.
  • A. Knutson, T. Tao, and C. Woodward, The honeycomb model of $\GL(n)$ tensor products, II: Puzzles determine facets of the L-R cone, to appear in J. Amer. Math. Soc., preprint.
  • B. Kostant and S. Kumar, The nil Hecke ring and cohomology of $G/P$ for a Kac-Moody group $G$, Adv. in Math. 62 (1986), 187--237.
  • A. Lascoux and M.-P. Schützenberger, ``Interpolation de Newton à plusieurs variables'' in Séminaire d'algèbre Paul Dubreil et Marie-Paule Malliavin, 36ème année (Paris, 1983--1984.), Lecture Notes in Math. 1146, Springer, Berlin, 1985, 161--175.
  • A. I. Molev and B. E. Sagan, A Littlewood-Richardson rule for factorial Schur functions, Trans. Amer. Math. Soc. 351 (1999), 4429--4443.
  • A. Okounkov, Quantum immanants and higher Capelli identities, Transform. Groups 1 (1996), 99--126.
  • S. Robinson, A Pieri-type formula for $H^*_T(\SL_n(\mathbbC)/B)$, J. Algebra 249 (2002), 38--58.