Related papers: Loop Grassmannians of quivers and affine quantum g…

Given a complex projective algebraic variety, write H(X) for its cohomology with complex coefficients and IH(X) for its Intersection cohomology. We first show that, under some fairly general conditions, the canonical map H(X)\to IH(X) is…

This article 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 naturally embedded in a torus. Then we…

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 We prove an unpublished result of D. Peterson which states that the quantum cohomology QH^*(G/P) of a flag variety is, up to localization, a…

The quantum cohomology of Grassmannians exhibits two symmetries related to the quantum product, namely a \Bbb {Z}/n action and an involution related to complex conjugation. We construct a new ring by dividing out these symmetries in an…

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 arXiv:math/0609796 and…

Let A be a finite abelian group. We set up an algebraic framework for studying A-equivariant complex-orientable cohomology theories in terms of a suitable kind of equivariant formal groups. We compute the equivariant cohomology of many…

There are two intriguing statements regarding the quantum cohomology of partial flag varieties. The first one relates quantum cohomology to the affinisation of Lie algebras and the homology of the affine Grassmannian, the second one…

Let g be a complex semisimple Lie algebra and let G' be the Langlands dual group. We give a description of the cohomology algebra of an arbitrary spherical Schubert variety in the loop Grassmannian for G' as a quotient of the form…

We give elementary proofs of the main theorems about (small) quantum cohomology of Grassmannians, including the quantum Giambelli and quantum Pieri formulas, the rim-hook algorithm, Siebert and Tian's presentation, and a recent theorem of…

The quantum cohomology ring of the Grassmannian is determined by the quantum Pieri rule for multiplying by Schubert classes indexed by row or column-shaped partitions. We provide a direct equivariant generalization of Postnikov's quantum…

We introduce and study a new mathematical structure in the generalised (quantum) cohomology theory for Grassmannians. Namely, we relate the Schubert calculus to a quantum integrable system known in the physics literature as the asymmetric…

Mirkovi\'c introduced the notion of loop Grassmannian for symmetric integer matrix $\kappa$. It is a two-step limit of the local projective space $Z_{\kappa}^{\alpha}$, which generalizes the usual Zastava for a simply laced group $G$. The…

We establish equivalences of derived categories of the following 3 categories: (1) Principal block of representations of the quantum at a root of 1; (2) G-equivariant coherent sheaves on the Springer resolution; (3) Perverse sheaves on the…

We extend ideas developed for the loop representation of quantum gravity to diffeomorphism-invariant gauge theories coupled to fermions. Let P -> Sigma be a principal G-bundle over space and let F be a vector bundle associated to P whose…

In this paper, we study the classical and quantum equivariant cohomology of Nakajima quiver varieties for a general quiver Q. Using a geometric R-matrix formalism, we construct a Hopf algebra Y_Q, the Yangian of Q, acting on the cohomology…

Let V be a symplectic vector space and LG be the Lagrangian Grassmannian which parametrizes maximal isotropic subspaces in V. We give a presentation for the (small) quantum cohomology ring QH^*(LG) and show that its multiplicative structure…

Let the vector bundle $\mathcal{E}$ be a deformation of the tangent bundle over the Grassmannian $G(k,n)$. We compute the ring structure of sheaf cohomology valued in exterior powers of $\mathcal{E}$, also known as the polymology. This is…

We study generalised differential structures $\Omega^1,d$ on an algebra $A$, where $A\tens A\to \Omega^1$ given by $a\tens b\to a d b$ need not be surjective. The finite set case corresponds to quivers with embedded digraphs, the Hopf…

We construct torus equivariant desingularizations of quiver Grassmannians for arbitrary nilpotent representations of an equioriented cycle quiver. We apply this to the computation of their torus equivariant cohomology.

We observe a general structure theorem for quantum cohomology rings, a non-homogeneous version of the usual cohomology ring encoding information about (almost holomorphic) rational curves. An application is the rigorous computation of the…