## 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 $$10.1$$).

### MSC:

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

### Keywords:

Gromov Witten invariants
Full Text:

### References:

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