## Duke Mathematical Journal

### Puzzles and (equivariant) cohomology of Grassmannians

#### 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

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

#### 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

#### References

• 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.