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

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

doi:10.1007/s10801-016-0697-3

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

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

School of Arts and Sciences, Mathematics

Research output: Contribution to journal › Article › peer-review

15 Scopus citations

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.

Original language	English (US)
Pages (from-to)	973-1007
Number of pages	35
Journal	Journal of Algebraic Combinatorics
Volume	44
Issue number	4
DOIs	https://doi.org/10.1007/s10801-016-0697-3
State	Published - Dec 1 2016

All Science Journal Classification (ASJC) codes

Algebra and Number Theory
Discrete Mathematics and Combinatorics

Keywords

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

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10.1007/s10801-016-0697-3

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title = "The puzzle conjecture for the cohomology of two-step flag manifolds",

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.",

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

author = "Buch, {Anders Skovsted} and Andrew Kresch and Kevin Purbhoo and Harry Tamvakis",

note = "Funding Information: 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). Publisher Copyright: {\textcopyright} 2016, Springer Science+Business Media New York.",

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AU - Buch, Anders Skovsted

AU - Kresch, Andrew

AU - Purbhoo, Kevin

AU - Tamvakis, Harry

N1 - Funding Information: 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). Publisher Copyright: © 2016, Springer Science+Business Media New York.

PY - 2016/12/1

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N2 - 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.

AB - 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.

KW - Gromov–Witten invariants

KW - Littlewood–Richardson rule

KW - Puzzle

KW - Quantum cohomology of Grassmannians

KW - Schubert calculus

KW - Two-step flag manifolds

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

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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