Crystal approach to affine Schubert calculus. (English) Zbl 1404.14057

Morse, Jennifer; Schilling, Anne

Int. Math. Res. Not. 2016, No. 8, 2239-2294 (2016).

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)