Abstract
We apply ideas from crystal theory to affine Schubert calculus and flag Gromov-Witten invariants. By defining operators on certain decompositions of elements in the type-$A$ affine Weyl group, we produce a crystal reflecting the internal structure of Specht modules associated to permutation diagrams. We show how this crystal framework can be applied to study the product of a Schur function with a $k$-Schur function. Consequently, we prove that a subclass of 3-point Gromov-Witten invariants of complete flag varieties for $\mathbb{C}^n$ enumerate the highest weight elements under these operators. Nous appliquons des idées provenant de la théorie des bases cristallines au calcul de Schubert affine et aux invariants de drapeaux de Gromov–Witten. Nous définissons des opérateurs sur certaines décompositions d’éléments de groupes de Weyl affines en type $A$ afin de construire une base cristalline encodant la structure interne des modules de Specht associés aux diagrammes de permutations. Nous montrons comment la structure de cristal permet d’étudier le produit d’une fonction de Schur avec une $k$-fonction de Schur. En conséquence, nous prouvons que la sous-classe des invariants de 3-points de Gromov–Witten d’une variété complète de drapeaux complets pour $\mathbb{C}^n$ énumère les éléments de poids maximaux pour ces opérateurs.
Highlights
Crystal theory [8, 9] and Schubert calculus convene beautifully in type gln
Developments in algebraic geometry and topology convert the problem into the computation of intersection numbers of certain subvarieties in the Grassmannian, which in turn is encoded by the structure constants of Schubert classes {σλ}λ⊂rect for the cohomology ring of the Grassmannian Gr(a, n)
Jennifer Morse and Anne Schilling the computation is made concrete with an isomorphism from H∗(Gr(a, n)) to the quotient of a polynomial ring under which Schubert classes correspond to Schur functions
Summary
Crystal theory [8, 9] and Schubert calculus convene beautifully in type gln. The crystal is the graph whose vertices are Young tableaux and whose edges are imposed by coplactic operators first introduced by Lascoux and Schutzenberger [21, 22]. We find that highest weight elements in the crystal correspond to certain coefficients in the product of a Schur times a k-Schur function. These are affine LR-coefficients (1) since s(wkμ) reduces to the. The multiplicative structure is defined by Squ ∗q Sqw = v q u, w, v d qd Sqw0v , where the structure constants are the 3-point Gromov–Witten invariants of genus 0, constants which count equivalence classes of certain rational curves in Fln. Peterson asserted that QH∗(G/P ) of a flag variety is a quotient of the homology H∗(GrG) of the affine Grassmannian up to localization (details carried out in [16]).
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