Abstract
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.
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The authors were supported in part by NSF Grants DMS-0906148 and DMS-1205351 (Buch), the Swiss National Science Foundation (Kresch), an NSERC discovery grant (Purbhoo), and NSF Grants DMS-0901341 and DMS-1303352 (Tamvakis).
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Buch, A.S., Kresch, A., Purbhoo, K. et al. The puzzle conjecture for the cohomology of two-step flag manifolds. J Algebr Comb 44, 973–1007 (2016). https://doi.org/10.1007/s10801-016-0697-3
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DOI: https://doi.org/10.1007/s10801-016-0697-3
Keywords
- Schubert calculus
- Two-step flag manifolds
- Puzzle
- Littlewood–Richardson rule
- Quantum cohomology of Grassmannians
- Gromov–Witten invariants