Journal of Algebraic Combinatorics

, Volume 44, Issue 4, pp 973–1007 | Cite as

The puzzle conjecture for the cohomology of two-step flag manifolds

  • Anders Skovsted Buch
  • Andrew Kresch
  • Kevin Purbhoo
  • Harry Tamvakis


We prove a conjecture of Knutson asserting that the Schubert structure constants of the cohomology ring of a two-step flag variety are equal to the number of puzzles with specified border labels that can be created using a list of eight puzzle pieces. As a consequence, we obtain a puzzle formula for the Gromov–Witten invariants defining the small quantum cohomology ring of a Grassmann variety of type A. The proof of the conjecture proceeds by showing that the puzzle formula defines an associative product on the cohomology ring of the two-step flag variety. It is based on an explicit bijection of gashed puzzles that is analogous to the jeu de taquin algorithm but more complicated.


Schubert calculus Two-step flag manifolds Puzzle Littlewood–Richardson rule Quantum cohomology of Grassmannians Gromov–Witten invariants 

Mathematics Subject Classification

Primary 05E05 Secondary 14N15 14M15 


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Anders Skovsted Buch
    • 1
  • Andrew Kresch
    • 2
  • Kevin Purbhoo
    • 3
  • Harry Tamvakis
    • 4
  1. 1.Department of MathematicsRutgers UniversityPiscatawayUSA
  2. 2.Institut für MathematikUniversität ZürichZurichSwitzerland
  3. 3.Combinatorics and Optimization DepartmentUniversity of WaterlooWaterlooCanada
  4. 4.Department of MathematicsUniversity of MarylandCollege ParkUSA

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