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


Buch, Anders Skovsted; Kresch, Andrew; Purbhoo, Kevin; Tamvakis, Harry (2016). The puzzle conjecture for the cohomology of two-step flag manifolds. Journal of Algebraic Combinatorics, 44(4):973-1007.

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.

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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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > Algebra and Number Theory
Physical Sciences > Discrete Mathematics and Combinatorics
Language:English
Date:July 2016
Deposited On:18 Jan 2017 12:41
Last Modified:01 Apr 2020 22:11
Publisher:Springer
ISSN:0925-9899
OA Status:Green
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1007/s10801-016-0697-3

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