Quantum hooks and mirror symmetry for flag varieties. (English) Zbl 1536.14029
Summary: Given a flag variety \(\mathrm{Fl}(n;r_1,\dots,r_\rho)\), there is natural ring morphism from the symmetric polynomial ring in \(r_1\) variables to the quantum cohomology of the flag variety. In this paper, we show that for a large class of partitions \(\lambda\), the image of \(s_\lambda\) under the ring homomorphism is a Schubert class which is described by partitioning \(\lambda\) into a quantum hook (or \(q\)-hook) and a tuple of smaller partitions. We use this result to show that the Plücker coordinate mirror of the flag variety describes quantum cohomology relations. This gives new insight into the structure of this superpotential, and the relation between superpotentials of flag varieties and those of Grassmannians (where the superpotential was introduced by B. R. Marsh and K. Rietsch [Adv. Math. 366, Article ID 107027, 131 p. (2020; Zbl 1453.14104)]).
MSC:
| 14J33 | Mirror symmetry (algebro-geometric aspects) |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
Citations:
Zbl 1453.14104References:
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