Related papers: Schubert Calculus on a Grassmann Algebra
A natural Hasse-Schmidt derivation on the exterior algebra of a free module realizes the (small quantum) cohomology ring of the grassmannian $G_k(\CC^n)$ as a ring of operators on the exterior algebra of a free module of rank $n$. Classical…
Let $T$ be a torus acting on $\CC^n$ in such a way that, for all $1\leq k\leq n$, the induced action on the grassmannian $G(k,n)$ has only isolated fixed points. This paper proposes a natural, elementary, explicit description of the…
The main classical result of Schubert calculus is that multiplication rules for the basis of Schubert cycles inside the cohomology ring of the Grassmannian $G(n,m)$ are the same as multiplication rules for the basis of Schur polynomials in…
The potential that generates the cohomology ring of the Grassmannian is given in terms of the elementary symmetric functions using the Waring formula that computes the power sum of roots of an algebraic equation in terms of its…
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…
Let $\ell, n$ be positive integers such that $\ell\geq n$. Let $\mathbb{G}_{n,\ell}$ be the Grassmannian which consists of the set of $n$-dimensional subspaces of $\mathbb{C}^{\ell}$. There is a $\mathbb{Z}$-graded algebra isomorphism…
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…
Imanishi, Jinzenji and Kuwata provided a recipe for computing Euler number of Grassmann manifold $G(k,N)$ using physical model and its path-integral [S.Imanishi, M.Jinzenji and K.Kuwata, Journal of Geometry and Physics, Volume 180, October…
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…
This paper studies, for a positive integer $m$, the subalgebra of the cohomology ring of the complex Grassmannians generated by the elements of degree at most $m$. We build in two ways upon a conjecture for the Hilbert series of this…
The maximal minors of a p by (m + p) matrix of univariate polynomials of degree n with indeterminate coefficients are themselves polynomials of degree np. The subalgebra generated by their coefficients is the coordinate ring of the quantum…
The {\em Schubert derivation} is a distinguished Hasse-Schmidt derivation on the exterior algebra of a free abelian group, encoding the formalism of Schubert calculus for all Grassmannians at once. The purpose of this paper is to extend the…
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 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…
We give formulas for the products of classes of Schubert varieties in the quantum cohomology rings of Grassmannians, in terms of the combinatorics of partitions and tableaux.
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 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…
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…
An important combinatorial result in equivariant cohomology and $K$-theory Schubert calculus is represented by the formulas of Billey and Graham-Willems for the localization of Schubert classes at torus fixed points. These formulas work…
The parabolic Kazhdan-Lusztig polynomials for Grassmannians can be computed by counting Dyck partitions. We "lift" this combinatorial formula to the corresponding category of singular Soergel bimodules to obtain bases of the Hom spaces…