An equivariant quantum Pieri rule for the Grassmannian on cylindric shapes. (English) Zbl 1498.14137
Summary: The quantum cohomology ring of the Grassmannian is determined by the quantum Pieri rule for multiplying by Schubert classes indexed by row or column-shaped partitions. We provide a direct equivariant generalization of Postnikov’s quantum Pieri rule for the Grassmannian in terms of cylindric shapes, complementing related work of V. Gorbounov and C. Korff [Adv. Math. 313, 282–356 (2017; Zbl 1386.14181)] in quantum integrable systems. The equivariant terms in our Graham-positive rule simply encode the positions of all possible addable boxes within one cylindric skew diagram. As such, unlike the earlier equivariant quantum Pieri rule of Y. Huang and C. Li [J. Algebra 441, 21–56 (2015; Zbl 1349.14173)] and known equivariant quantum Littlewood-Richardson rules, our formula does not require any calculations in a different Grassmannian or two-step flag variety.
MSC:
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 14N15 | Classical problems, Schubert calculus |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
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