Let G be a simple and simply-connected complex algebraic group, P \subset G a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology QH^*(G/P) of a flag variety is, up to localization, a quotient of the homology H_*(Gr_G) of the affine Grassmannian \Gr_G of G. As a consequence, all three-point genus zero Gromov-Witten invariants of $G/P$ are identified with homology Schubert structure constants of H_*(Gr_G), establishing the equivalence of the quantum and homology affine Schubert calculi. 
For the case G = B, we use the Mihalcea's equivariant quantum Chevalley formula for QH^*(G/B), together with relationships between the quantum Bruhat graph of Brenti, Fomin and Postnikov and the Bruhat order on the affine Weyl group. As byproducts we obtain formulae for affine Schubert homology classes in terms of quantum Schubert polynomials. We give some applications in quantum cohomology. 
Our main results extend to the torus-equivariant setting.

Quantum cohomology of G/P and homology of affine Grassmannian

These are lecture notes intended to supplement my second lecture at the Current Developments in Mathematics conference in 2014 In the first half of article, we give an introduction to the totally nonnegative Grassmannian together with a survey of some more recent work In the second half of the article, we give a definition of a Grassmann polytope motivated by work of physicists on the amplituhedron We propose to use Schubert calculus and canonical bases to replace linear algebra and convexity in the theory of polytopes

Totally nonnegative Grassmannian and Grassmann polytopes

This book is an exposition of the current state of research of affine Schubert calculus and $k$-Schur functions. This text is based on a series of lectures given at a workshop titled "Affine Schubert Calculus" that took place in July 2010 at the Fields Institute in Toronto, Ontario. The story of this research is told in three parts: 1. Primer on $k$-Schur Functions 2. Stanley symmetric functions and Peterson algebras 3. Affine Schubert calculus

$k$-Schur functions and affine Schubert calculus

Quantum integrability and generalised quantum Schubert calculus

We study the quantum cohomology of (co)minuscule homogeneous varieties under a unified perspective. We show that three points Gromov–Witten invariants can always be interpreted as classical intersection numbers on auxiliary varieties. Our main combinatorial tools are certain quivers, in terms of which we obtain a quantum Chevalley formula and a higher quantum Poincare duality. In particular, we compute the quantum cohomology of the two exceptional minuscule homogeneous varieties.

/pdf/quantum-cohomology-of-minuscule-homogeneous-spaces-4xwa0up8l3.pdf

Quantum Cohomology of Minuscule Homogeneous Spaces

We study the quantum cohomology of quasi-minuscule and quasi-cominuscule homogeneous spaces. The product of any two Schubert cells does not involve powers of the quantum parameter higher than 2. With the help of the quantum to classical principle we give presentations of the quantum cohomology algebras. These algebras are semi-simple for adjoint non coadjoint varieties and some properties of the induced strange duality are shown.

On the quantum cohomology of adjoint varieties

Let $X$ be a rational homogeneous space and let $QH^*(X)_{loc}^\times$ be the group of invertible elements in the small quantum cohomology ring of $X$ localised in the quantum parameters. We generalise results of \cite{cmp2} and realise explicitly the map $\pi_1({\rm Aut}(X))\to QH^*(X)_{loc}^\times$ described in \cite{seidel}. We even prove that this map is an embedding and realise it in the equivariant quantum cohomology ring $QH^*_T(X)_{loc}^\times$. We give explicit formulas for the product by these elements. The proof relies on a generalisation, to a quotient of the equivariant homology ring of the affine Grassmannian, of a formula proved by Peter Magyar \cite{Magyar}. It also uses Peterson's unpublished result \cite{Peterson} --- recently proved by Lam and Shimozono in \cite{Lam-Shi} --- on the comparison between the equivariant homology ring of the affine Grassmannian and the equivariant quantum cohomology ring.

pdf/affine-symmetries-of-the-equivariant-quantum-cohomology-ring-33cz9u485q.pdf

Affine symmetries of the equivariant quantum cohomology ring of rational homogeneous spaces

We show that for any minuscule or cominuscule homogeneous space X, the Gromov–Witten varieties of degree d curves passing through three general points of X are rational or empty for any d. Applying techniques of Buch and Mihalcea BM to constructions of the authors together with Manivel [5], we deduce that the equivariant K-theoretic three points Gromov–Witten invariants are equal to classical equivariant K-theoretic invariants on auxilliary spaces again homogeneous under Aut(X).

Rationality of some Gromov-Witten varieties and application to quantum K-theory

We prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, specialized at q=1, is semisimple. This implies that complex conjugation defines an algebra automorphism of the quantum cohomology ring localized at the quantum parameter. We check that this involution coincides with the strange duality defined in a previous paper. We deduce Vafa-Intriligator type formulas for the Gromov-Witten invariants.

Quantum cohomology of minuscule homogeneous spaces III : semi-simplicity and consequences

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