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

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

Research output: Contribution to journalArticlepeer-review

15 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)973-1007
Number of pages35
JournalJournal of Algebraic Combinatorics
Issue number4
StatePublished - Dec 1 2016

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Discrete Mathematics and Combinatorics


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


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