# Crystal approach to affine Schubert calculus

@article{Morse2016CrystalAT,
title={Crystal approach to affine Schubert calculus},
author={Jennifer Morse and Anne Schilling},
journal={International Mathematics Research Notices},
year={2016},
volume={2016},
pages={2239-2294}
}
• Published 1 August 2014
• Mathematics
• International Mathematics Research Notices
Author(s): Morse, Jennifer; Schilling, Anne | Abstract: 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…

## Figures from this paper

k-Schur expansions of Catalan functions
• Mathematics
• 2020
Highest weight crystals for Schur Q-functions
• Mathematics
• 2021
Work of Grantcharov et al. develops a theory of abstract crystals for the queer Lie superalgebra qn. Such qn-crystals form a monoidal category in which the connected normal objects have unique
Combinatorial description of the cohomology of the affine flag variety
• Seung Jin Lee
• Mathematics
Discrete Mathematics & Theoretical Computer Science
• 2020
International audience We construct the affine version of the Fomin-Kirillov algebra, called the affine FK algebra, to investigatethe combinatorics of affine Schubert calculus for typeA. We
Bumping operators and insertion algorithms for queer supercrystals
Results of Morse and Schilling show that the set of increasing factorizations of reduced words for a permutation is naturally a crystal for the general linear Lie algebra. Hiroshima has recently
Schur P-positivity and involution Stanley symmetric functions
• Mathematics
• 2017
The involution Stanley symmetric functions $\hat{F}_y$ are the stable limits of the analogues of Schubert polynomials for the orbits of the orthogonal group in the flag variety. These symmetric
Applying parabolic Peterson: affine algebras and the quantum cohomology of the Grassmannian
• Mathematics
Journal of Combinatorics
• 2019
The Peterson isomorphism relates the homology of the affine Grassmannian to the quantum cohomology of any flag variety. In the case of a partial flag, Peterson's map is only a surjection, and one
CRYSTAL STRUCTURES FOR SYMMETRIC GROTHENDIECK POLYNOMIALS
• Mathematics
• 2018
The symmetric Grothendieck polynomials representing Schubert classes in the K theory of Grassmannians are generating functions for semistandard set-valued tableaux. We construct a type A n crystal
Shifted tableaux crystals
• Mathematics
• 2017
We introduce coplactic raising and lowering operators $E'_i$, $F'_i$, $E_i$, and $F_i$ on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each
A Demazure crystal construction for Schubert polynomials
• Mathematics
• 2018
Stanley symmetric functions are the stable limits of Schubert polynomials. In this paper, we show that, conversely, Schubert polynomials are Demazure truncations of Stanley symmetric functions. This
Alcove random walks, k-Schur functions and the minimal boundary of the k-bounded partition poset
• Mathematics
Algebraic Combinatorics
• 2020
We use k-Schur functions to get the minimal boundary of the k-bounded partition poset. This permits to describe the central random walks on affine Grassmannian elements of type A and yields a

## References

SHOWING 1-10 OF 92 REFERENCES
Schur Times Schubert via the Fomin-Kirillov Algebra
• Mathematics
Electron. J. Comb.
• 2014
An algebro-combinatorial proof of the nonnegativity of the Gromov-Witten invariants in these cases is provided, and combinatorial expressions for these coefficients are presented.
Affine approach to quantum Schubert calculus
This paper presents a formula for products of Schubert classes in the quantum cohomology ring of the Grassmannian. We introduce a generalization of Schur symmetric polynomials for shapes that are
Some Combinatorial Properties of Schubert Polynomials
• Mathematics
• 1993
AbstractSchubert polynomials were introduced by Bernstein et al. and Demazure, and were extensively developed by Lascoux, Schützenberger, Macdonald, and others. We give an explicit combinatorial
Schubert Polynomials and $k$-Schur Functions
• Mathematics
Electron. J. Comb.
• 2014
It is shown that the multiplication of a SchUbert polynomial of finite type $A$ by a Schur function, which is referred to as Schubert vs. Schur problem, can be understood from the multiplication in the space of dual $k$-Schur functions.
A Unified Approach to Combinatorial Formulas for Schubert Polynomials
Schubert polynomials were introduced in the context of the geometry of flag varieties. This paper investigates some of the connections not yet understood between several combinatorial structures for
QUANTUM COHOMOLOGY AND THE k-SCHUR BASIS
• Mathematics
• 2007
We prove that structure constants related to Hecke algebras at roots of unity are special cases of k-Littlewood-Richardson coefficients associated to a product of k-Schur functions. As a consequence,
Quantum Schubert polynomials
• Mathematics
• 1997
where In is the ideal generated by symmetric polynomials in x1,... ,xn without constant term. Another, geometric, description of the cohomology ring of the flag manifold is based on the decomposition
Puzzles and (equivariant) cohomology of Grassmannians
• Mathematics
• 2001
We generalize our puzzle formula for ordinary Schubert calculus on Grassmannians, to a formula for the T-equivariant Schubert calculus. The structure constants to be calculated are polynomials in
k-Schur Functions and Affine Schubert Calculus
• Mathematics
• 2014
Author(s): Lam, T; Morse, J; Shimozono, M; Lapointe, L; Schilling, A; Zabrocki, M | Abstract: This book is an exposition of the current state of research of affine Schubert calculus and $k$-Schur
Quantum cohomology of G/P and homology of affine Grassmannian
• Mathematics
• 2007
Let G be a simple and simply-connected complex algebraic group, P ⊂ G a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology QH*(G/P) of a flag