Quantum cohomology of Grassmannians modulo symmetries
Harald Hengelbrock
备注：
11 pages, presented at the 2002 Fall AMS Eastern Section Meeting on ring structures in Schubert calculus,Northeastern University, Boston,October 5-6, 2002
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 ideal theoretic way and analyze its structure, which is shown to control the sum of all coefficients appearing in the product of cohomology classes. We derive a combinatorial formula for the sum of all Littlewood-Richardson coefficients appearing in the expansion of a product of two Schur polynomials.
We give a series of successively weaker conjectures on the cohomology ring of the Grassmannian, starting with the Hilbert series of a certain natural filtration.
We give positive formulas for the restriction of a Schubert Class to a T-fixed point in the equivariant K-theory and equivariant cohomology of the Lagrangian Grassmannian. Our formulas rely on a result of Ghorpade-Raghavan, which gives an equivariant Grobner degeneration of a Schubert variety in the neighborhood of a T-fixed point of the Lagrangian Grassmannian.
We find presentations by generators and relations for the equivariant quantum cohomology of the Grassmannian. For these presentations, we also find determinantal formulae for the equivariant quantum Schubert classes. To prove this, we use the theory of factorial Schur functions and a characterization of the equivariant quantum cohomology ring.
We give positive formulas for the restriction of a Schubert Class to a T-fixed point in the equivariant K-theory and equivariant cohomology of the Grassmannian. Our formulas rely on a result of Kodiyalam-Raghavan and Kreiman-Lakshmibai, which gives an equivariant Grobner degeneration of a Schubert variety in the neighborhood of a T-fixed point of the Grassmannian.
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 needs to quotient by a suitable ideal on the affine side to map isomorphically onto the quantum cohomology. We provide a detailed exposition of this parabolic Peterson isomorphism in the case of the Grassmannian of m-planes in complex n-space, including an explicit recipe for doing quantum Schubert calculus in terms of the appropriate subset of non-commutative k-Schur functions. As an application, we …
The odd symplectic Grassmannian $\mathrm{IG}:=\mathrm{IG}(k, 2n+1)$ parametrizes $k$ dimensional subspaces of $\mathbb{C}^{2n+1}$ which are isotropic with respect to a general (necessarily degenerate) symplectic form. The odd symplectic group acts on $\mathrm{IG}$ with two orbits, and $\mathrm{IG}$ is itself a smooth Schubert variety in the submaximal isotropic Grassmannian $\mathrm{IG}(k, 2n+2)$. We use the technique of curve neighborhoods to prove a Chevalley formula in the equivariant quantum cohomology of $\mathrm{IG}$, i.e. a formula to multiply a Schubert class by the Schubert divisor cl…
We prove that if X is a Grassmannian of type A, then the Schubert basis of the (small) quantum cohomology ring QH(X) is the only homogeneous deformation of the Schubert basis of the ordinary cohomology ring of X that multiplies with non-negative structure constants. This implies that the (three point, genus zero) Gromov-Witten invariants of X are uniquely determined by Witten's presentation of QH(X) and the fact that they are non-negative. We conjecture that the same is true for any flag variety X = G/P of simply laced Lie type. For the variety GL(n)/B of complete flags, this conjecture is equ…
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 Fulton and Woodward about the minimal q-power which appears in a product of two Schubert classes.
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 …
Odd symplectic Grassmannians are a generalization of symplectic Grassmannians to odd-dimensional spaces. Here we compute the classical and quantum cohomology of the odd symplectic Grassmannian of lines. Although these varieties are non homogeneous, we obtain Pieri and Giambelli formulas that are very similar to the symplectic case. We notice that their quantum cohomology is semi-simple, which enables us to check Dubrovin's conjecture for this case.
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, both the 3-point Gromov-Witten invariants appearing in the quantum cohomology of the Grassmannian, and the fusion coefficients for the WZW conformal field theories associated to \hat{su}(\ell) are shown to be k-Littlewood Richardson coefficients. From this, Mark Shimozono conjectured that the k-Schur functions form the Schubert basis for the homology of the loop Grassmannian, whereas k-Schur copro…
Let $LG_n$ denote the Lagrangian Grassmannian parametrizing maximal isotropic (Lagrangian) subspaces of a fixed symplectic vector space of dimension $2n.$ For each strict partition $λ=(λ_1,...,λ_k)$ with $λ_1\leq n$ there is a Schubert variety $X(λ).$ Let $T$ denote a maximal torus of the symplectic group acting on $LG_n.$ Consider the $T$-equivariant cohomology of $LG_n$ and the $T$-equivariant fundamental class $σ(λ)$ of $X(λ).$ The main result of the present paper is an explicit formula for the restriction of the class $σ(λ)$ to any torus fixed point. The formula is written in terms of fact…
The Drinfel'd Lagrangian Grassmannian compactifies the space of algebraic maps of fixed degree from the projective line into the Lagrangian Grassmannian. It has a natural projective embedding arising from the canonical embedding of the Lagrangian Grassmannian. We show that the defining ideal of any Schubert subvariety of the Drinfel'd Lagrangian Grassmannian is generated by polynomials which give a straightening law on an ordered set. Consequentially, any such subvariety is Cohen-Macaulay and Koszul. The Hilbert function is computed from the straightening law, leading to a new derivation of ce…
We compute the small cohomology ring of the Cayley Grassmannian, that parametrizes four-dimensional subalgebras of the complexified octonions. We show that all the Gromov-Witten invariants in the multiplication table of the Schubert classes are non negative and deduce Golyshev's conjecture O holds true for this variety. We also check that the quantum cohomology is semisimple and that there exists, as predicted by Dubrovin's conjecture, an exceptional collection of maximal length in the derived category.
A driving question in (quantum) cohomology of flag varieties is to find non-recursive, positive combinatorial formulas for expressing the product of two classes in a particularly nice basis, called the Schubert basis. Bertram, Ciocan-Fontanine and Fulton provided a way to compute quantum products of Schubert classes in the Grassmannian of k-planes in complex n-space by doing classical multiplication and then applying a combinatorial rim hook rule which yields the quantum parameter. In this paper, we provide a generalization of this rim hook rule to the setting in which there is also an action …
For a Fano manifold M, complex conjugation defines a real involution on the quantum cohomology ring. For the Grassmannian we identify this involution with an explicit transformation on Schubert classes defined over the integers. It is a composition of a power of the cyclic action recently defined by Agnihotri and Woodward with Poincare duality. The existence of the involution has been observed independently and from a different point of view by Postnikov (math.CO/0205165).
We give a proof of a result of D. Peterson's identifying the quantum cohomology ring of a Grassmannian with the reduced coordinate ring of a certain subvariety of $GL_n$. The totally positive part of this subvariety is then constructed and we give closed formulas for the values of the Schubert basis elements on the totally positive points. We then use the developed methods to give a new proof of a formula of Vafa and Intriligator and Bertram for the structure constants (Gromov--Witten invariants). Finally, we use the positivity of these Gromov--Witten invariants to prove certain inequalities f…
It was proved by Ginzburg and Mirkovic-Vilonen that the $G(O)$-equivariant perverse sheaves on the affine grassmannian of a connected reductive group $G$ form a tensor category equivalent to the tensor category of finite dimensional representations of the dual group $G^\vee$. The proof use the Tannakian formalism. The purpose of this paper is to construct explicitely the action of $G^\vee$ on the global cohomology of a perverse sheaf.
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 is determined by the ring of (Q^~)-polynomials. We formulate a "quantum Schubert calculus" which includes quantum Pieri and Giambelli formulas, as well as algorithms for computing the structure constants appearing in the quantum product of Schubert classes.
Let V be a vector space with a nondegenerate symmetric form and OG be the orthogonal Grassmannian which parametrizes maximal isotropic subspaces in V. We give a presentation for the (small) quantum cohomology ring QH^*(OG) and show that its product structure is determined by the ring of (P~)-polynomials. A "quantum Schubert calculus" is formulated, which includes quantum Pieri and Giambelli formulas, as well as algorithms for computing Gromov-Witten invariants. As an application, we show that the table of 3-point, genus zero Gromov-Witten invariants for OG coincides with that for a correspondi…
We introduce the notion of alternate product of Frobenius manifolds and we give, after [math.AG/0610265], an interpretation of the Frobenius manifold structure canonically attached to the quantum cohomology of G(r,n+1) in terms of alternate products. We also investigate the relationship with the alternate Thom-Sebastiani product of Laurent polynomials.
We study the Schubert calculus of the affine Grassmannian Gr of the symplectic group. The integral homology and cohomology rings of Gr are identified with dual Hopf algebras of symmetric functions, defined in terms of Schur's P and Q-functions. An explicit combinatorial description is obtained for the Schubert basis of the cohomology of Gr, and this is extended to a definition of the affine type C Stanley symmetric functions. A homology Pieri rule is also given for the product of a special Schubert class with an arbitrary one.
For an acyclic quiver, we establish a connection between the cohomology of quiver Grassmannians and the dual canonical bases of the algebra $U_q^-(\mathfrak{g})$, where $U_q^-(\mathfrak{g})$ is the negative half of the quantized enveloping algebra associated with the quiver. In order to achieve this goal, we study the cohomology of quiver Grassmannians by Lusztig's category. As a consequence, we describe explicitly the Poincaré polynomials of rigid quiver Grassmannians in terms of the coefficients of dual canonical bases, which are viewed as elements of quantum shuffle algebras. By this result…
In this note we describe the cohomology ring of the Grassmannian of $k$-planes in $n$-dimensional complex vector space as an $\mathrm{GL}_n$-module. We give explicit formulas for the operators of its principal $\mathfrak{sl}_2$-triple. It is proved that one of these operators corresponds to the shifted cohomology degree operator and the second operator coincides with the Lefschetz map on cohomology (as in the hard Lefschetz theorem). We check that the cohomology ring of the complex Grassmannian as a $\mathrm{GL}_n$-representation is isomorphic to the $k$-th exterior power of the standard $n$-d…
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 the first part of a project studying the quantum sheaf cohomology of Grassmannians with deformations of the tangent bundle, a generalization of ordinary quantum cohomology rings of Grassmannians. A companion physics paper [arXiv:1512.08586] describes physical aspects of the theory, including a conjecture for the quantum sheaf cohomology ring, and numerous exa…
We show that the cohomology ring of a quiver Grassmannian asssociated with a rigid quiver representation has property (S): there is no odd cohomology and the cycle map is an isomorphism; moreover, its Chow ring admits explicit generators defined over any field. From this we deduce the polynomial point count property. By restricting the quiver to finite or affine type, we are able to show a much stronger assertion: namely, that a quiver Grassmannian associated to an indecomposable (not necessarily rigid) representation admits a cellular decomposition. As a corollary, we establish a cellular dec…
We verify in an elementary way a result of Peterson for the maximal orthogonal and Lagrangian Grassmannians, and then find Vafa-Intriligator type formulas which compute their 3-point, genus zero Gromov-Witten invariants. Finally we study total positivity of the related Peterson varieties and show that Rietsch's conjecture about the total positivity holds for these cases.
In [Rie08], the second author defined a Landau-Ginzburg model for homogeneous spaces G/P, as a regular function on an affine subvariety of the Langlands dual group. In this paper, we reformulate this LG-model (X^,W_t) in the case of the Lagrangian Grassmannian LG(m) as a rational function on a Langlands dual orthogonal Grassmannian, in the spirit of work by R. Marsh and the second author [MR12] for type A Grassmannians. This LG model has some very interesting features, which are not visible in the type A case, to do with the non-triviality of Langlands duality. We also formulate a conjecture r…
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 {y_{i+1} - y_i}; they were shown (abstractly) to have positive coefficients in [Graham] math.AG/9908172. Our formula is the first to be manifestly positive in this sense. In particular this gives a new and self-contained proof of the ordinary puzzle formula, by an induction backwards from the "most equivariant" case. The proof of the formula is mostly combinatorial, but requires no prior combinatorics…
In this paper, we prove a generalization of Kempf-Laksov formula for the degeneracy loci classes in even infinitesimal cohomology theories of the Grassmannian bundle and the Lagrangian Grassmannian bundle.
This chapter concerns edge labeled Young tableaux, introduced by H. Thomas and the third author. It is used to model equivariant Schubert calculus of Grassmannians. We survey results, problems, conjectures, together with their influences from combinatorics, algebraic and symplectic geometry, linear algebra, and computational complexity. We report on a new shifted analogue of edge labeled tableaux. Conjecturally, this gives a Littlewood-Richardson rule for the structure constants of the D. Anderson-W. Fulton ring, which is related to the equivariant cohomology of isotropic Grassmannians.
In this article, we make a close analysis on quantum multiplication operators on the quantum cohomology rings of Lagrangian and orthogonal Grassmannians, and give an explicit description on all simultaneous eigenvectors and the corresponding eigenvalues for these operators. As a result, we show that Conjecture $\mathcal{O}$ of Galkin, Golyshev and Iritani holds for these manifolds.
The occurrence and frequency of a phenomenon of resonance (namely the coalescence of some Dubrovin canonical coordinates) in the locus of Small Quantum Cohomology of complex Grassmannians is studied. It is shown that surprisingly this frequency is strictly subordinate and highly influenced by the distribution of prime numbers. Two equivalent formulations of the Riemann Hypothesis are given in terms of numbers of complex Grassmannians without coalescence: the former as a constraint on the disposition of singularities of the analytic continuation of the Dirichlet series associated to the sequenc…
The Poincare duality of classical cohomology and the extension of this duality to quantum cohomology endows these rings with the structure of a Frobenius algebra. Any such algebra possesses a canonical ``characteristic element;'' in the classical case this is the Euler class, and in the quantum case this is a deformation of the classical Euler class which we call the ``quantum Euler class.'' We prove that the characteristic element of a Frobenius algebra A is a unit if and only if A is semisimple, and then apply this result to the cases of the quantum cohomology of the finite complex Grassmann…
We prove the conjectural relation between the Stokes matrix for the quantum cohomology and an exceptional collection generating the derived category of coherent sheaves in the case of the Grassmannian. The proof is based on the relation between the quantum cohomology of the Grassmannian and that of the projective space.
In this paper, we study the T_w-equivariant cohomology of the weighted Grassmannians wGr(d,n) introduced by Corti-Reid where T_w is the n-dimensional torus that naturally acts on wGr(d,n). We introduce the equivariant weighted Schubert classes and, after we show that they form a basis of the equivariant cohomology, we give an explicit formula for the structure constants with respect to this Schubert basis. We also find a linearly independent subset {wu_1,...,wu_n} of Lie(T_w)^* such that those structure constants are polynomials in wu_i's with non-negative coefficients, up to a permutation on …
Let G be a reductive group; in this note we give an interpretation of the dynamical Weyl group of of the Langlands dual group $\check{G}$ defined by Etingof and Varchenko in terms of the geometry of the affine Grassmannian Gr of G. In this interpretation the dynamical parameters of Etingof and Varchenko correspond to equivariant parameters with respect to certain natural torus acting on Gr. We also present a conjectural generalization of our results to the case of affine Kac-Moody groups.
The integral singular cohomology ring of the Grassmann variety parametrizing $r$-dimensional subspaces in the $n$-dimensional complex vector space is naturally an irreducible representation of the Lie algebra of all the $n\times n$ matrices with integral entries. Using the notion of Schubert derivation, a distinguished Hasse-Schmidt derivation on an exterior algebra, we describe explicitly such a representation, indicating its relationship with the celebrated bosonic vertex representation of the Lie algebra of infinite matrices due to Date, Jimbo, Kashiwara and Miwa.
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 injective. Now let Gr = G((z))/G[[z]] be the loop Grassmannian for a complex semisimple group G, and let X be the closure of a G[[z]]-orbit in Gr. We prove, using the general result above, a conjecture of D. Peterson describing the cohomology algebra H(X) in terms of the centralizer of the principal nilpotent in the Langlands dual of Lie(G). In the last sectio…
We introduce two families of symmetric functions with an extra parameter t that specialize to Schubert representatives for cohomology and homology of the affine Grassmannian when t = 1. The families are defined by a statistic on combinatorial objects associated to the type-A affine Weyl group and their transition matrix with Hall-Littlewood polynomials is t-positive. We conjecture that one family is the set of k-atoms.
The rim-hook rule for quantum cohomology of the Grassmannian allows one to reduce quantum calculations to classical calculations in the cohomology of the Grassmannian. We use the Abelian/non-Abelian correspondence for cohomology to prove a rim-hook removal rule for the cohomology of quiver flag varieties. Quiver flag varieties are generalisations of type A flag varieties; this result is new even in the flag case. This gives an effective way of computing products in their cohomology, reducing computations to that in the cohomology ring of the Grassmannian. We then prove a quantum rim-hook rule …
In this paper, we introduce a family of hyper-elliptic curves. For this family we compute the matrix of the divided Frobenius and we obtain general formulas. We use a recent results of Huyghe-Wach : the divided Frobenius coincides with the morphism built by Deligne-Illusie in 1987.
We prove a Chevalley formula for the equivariant quantum multiplication of two Schubert classes in the homogeneous variety X=G/P. As in the case when X is a Grassmannian, studied by the author in a previous paper, this formula implies an effective algorithm to compute the structure constants of the equivariant quantum cohomology algebra of X.
We study the three point genus zero Gromov-Witten invariants on the Grassmannians which parametrize non-maximal isotropic subspaces in a vector space equipped with a nondegenerate symmetric or skew-symmetric form. We establish Pieri rules for the classical cohomology and the small quantum cohomology ring of these varieties, which give a combinatorial formula for the product of any Schubert class with certain special Schubert classes. We also give presentations of these rings, with integer coefficients, in terms of special Schubert class generators and relations.
We prove that the geometric Satake correspondence admits quantum corrections for minuscule Grassmannians of Dynkin types $A$ and $D$. We find, as a corollary, that the quantum connection of a spinor variety $OG(n,2n)$ can be obtained as the half-spinorial representation of that of the quadric $Q_{2n-2}$. We view the (quantum) cohomology of these Grassmannians as endowed simultaneously with two structures, one of a module over the algebra of symmetric functions, and the other, of a module over the Langlands dual Lie algebra, and investigate the interaction between the two. In particular, we stu…
The subject of this paper is the big quantum cohomology rings of symplectic isotropic Grassmannians $\text{IG}(2, 2n)$. We show that these rings are regular. In particular, by "generic smoothness", we obtain a conceptual proof of generic semisimplicity of the big quantum cohomology for $\text{IG}(2, 2n)$. Further, by a general result of Claus Hertling, the regularity of these rings implies that they have a description in terms of isolated hypersurface singularities, which we show in this case to be of type $A_{n-1}$. By the homological mirror symmetry conjecture, these results suggest the exis…
