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Crystal approach to affine Schubert calculus. (English) Zbl 1404.14057
Summary: We apply crystal theory to affine Schubert calculus, Gromov-Witten invariants for the complete flag manifold, and the positroid stratification of the positive Grassmannian. We introduce operators on decompositions of elements in the type-$$A$$ affine Weyl group and produce a crystal reflecting the internal structure of the generalized Young modules whose Frobenius image is represented by stable Schubert polynomials. We apply the crystal framework to products of a Schur function with a $$k$$-Schur function, consequently proving that a subclass of three-point Gromov-Witten invariants of complete flag varieties for $$\mathbb C^n$$ enumerate the highest weight elements under these operators. Included in this class are the Schubert structure constants in the (quantum) product of a Schubert polynomial with a Schur function $$s_\lambda$$ for all $$|\lambda^\vee| < n$$. Another by-product gives a highest weight formulation for various fusion coefficients of the Verlinde algebra and for the Schubert decomposition of certain positroid classes.

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