The puzzle conjecture for the cohomology of two-step flag manifolds
The puzzle conjecture for the cohomology of two-step flag manifolds
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两步旗流形上同调的谜题猜想
DOI：
10.1007/s10801-016-0697-3
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发表时间：
2014
影响因子：
0.8
通讯作者：
Harry Tamvakis
中科院分区：
文献类型：
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作者：
A. Buch;A. Kresch;K. Purbhoo;Harry Tamvakis
我们证明了克努森（Knutson）的一个猜想，该猜想断言两步旗簇的上同调环的舒伯特（Schubert）结构常数等于使用八个拼图块的列表可以创建的具有指定边界标签的拼图数量。因此，我们得到了一个用于定义A类格拉斯曼（Grassmann）簇的小量子上同调环的格罗莫夫 - 威滕（Gromov - Witten）不变量的拼图公式。该猜想的证明通过表明拼图公式在两步旗簇的上同调环上定义了一个结合乘积来进行。它基于一种有缺口拼图的显式双射，这种双射类似于跳棋算法（jeu de taquin algorithm），但更为复杂。
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.