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On the quantum product of Schubert classes. (English) Zbl 1081.14076
The paper under review investigates the small quantum cohomology ring of general flag varieties \(G/P\). The quantum product deforms the classical cup product by adding contributions from the count of degree \(d\) rational curves on \(G/P\) with prescribed incidence conditions. The authors determine the smallest power of the quantum parameter that can occur in a product of two Schubert classes. This minimal degree is described combinatorially in terms of the Bruhat ordering, and geometrically by the \(11\) equivalent conditions of theorem \(9.1\) in the paper. The classical Chevalley’s formula computes the the product of two Schubert classes, one of of them being of codimension \(1\). The methods of this paper allow for a proof of the quantum version of this formula (theorem \(10.1\)).

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14M15 Grassmannians, Schubert varieties, flag manifolds
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