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

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...

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Bibliographic Details
Published in:Journal of algebraic combinatorics 2016-12, Vol.44 (4), p.973-1007
Main Authors: Buch, Anders Skovsted, Kresch, Andrew, Purbhoo, Kevin, Tamvakis, Harry
Format: Article
Language:English
Subjects:
Combinatorics
Computer Science
Convex and Discrete Geometry
Group Theory and Generalizations
Lattices
Mathematics
Mathematics and Statistics
Order
Ordered Algebraic Structures
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Summary: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.
ISSN:0925-9899
1572-9192
DOI:10.1007/s10801-016-0697-3