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arXiv · 1401.1725

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

Anders Skovsted Buch·Andrew Kresch·Kevin Purbhoo·Harry Tamvakis
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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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Anders Skovsted Buch, Andrew Kresch, Kevin Purbhoo, Harry Tamvakis. 2014-01-08. The puzzle conjecture for the cohomology of two-step flag manifolds. https://arxiv.org/abs/1401.1725

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