Related papers: On the quantum product of Schubert classes
We prove a Seidel product formula for the torus-equivariant quantum $K$-theory of a generalized flag variety $G/P.$ This is a natural generalization of the corresponding results by Buch, Chaput, and Perrin for the cominuscule flag…
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…
We give an explicit combinatorial Chevalley-type formula for the equivariant K-theory of generalized flag varieties G/P which is a direct generalization of the classical Chevalley formula. Our formula implies a simple combinatorial model…
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…
Consider a partial flag variety $X$ which is not a grassmaninan. Consider also its cohomology ring ${\rm H}^*(X,\ZZ)$ endowed with the base formed by the Poincar\'e dual classes of the Schubert varieties. In \cite{Richmond:recursion}, E.…
In this paper, we give a rule for the multiplication of a Schubert class by a tautological class in the (small) quantum cohomology ring of the flag manifold. As an intermediate step, we establish a formula for the multiplication of a…
We obtain an algorithm computing the Chern-Schwartz-MacPherson (CSM) classes of Schubert cells in a generalized flag manifold G/B. In analogy to how the ordinary divided difference operators act on Schubert classes, each CSM class of a…
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…
We describe the inequalities on the possible eigenvalues of products of unitary matrices in terms of quantum Schubert calculus. Related problems are the existence of flat connections on the punctured two-sphere with prescribed holonomies,…
We study multiplicative properties of the (quantum) dual canonical basis B* associated to a semi-simple complex Lie group G. We provide a subset D of B* such that the following property holds : if two elements b, b' in B* q-commute and if…
This paper works out the versions of the classical Giambelli and Pieri formulas in the context of quantum cohomology of a complex Grassmannian.
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…
We aim in this manuscript to describe a specific notion of geometric positivity that manifests in cohomology rings associated to the flag variety $G/B$ and, in some cases, to subvarieties of $G/B$. We offer an exposition on the the…
Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a "Giambelli formula" expressing…
The purpose of this paper is to prove a Pieri-type multiplication formula for quantum Grothendieck polynomials, which was conjectured by Lenart-Maeno. This formula would enable us to compute explicitly the quantum product of two arbitrary…
We find presentations by generators and relations for the equivariant quantum cohomology rings of the maximal isotropic Grassmannians of types B,C and D, and we find polynomial representatives for the Schubert classes in these rings. These…
Let G be a compact connected Lie group and H, the centralizer of a one-parameter subgroup in G. Combining the ideas of Bott-Samelson resulotions of Schubert varieties and the enumerative formula on a twisted products of 2-spheres obatained…
A conjecture of Buch-Chaput-Perrin asserts that the two-pointed curve neighborhood corresponding to a quantum product of Seidel type is an explicitly given Schubert variety. We prove this conjecture for flag varieties in type A.
We construct a Pl\"ucker coordinate superpotential $\mathcal{F}_-$ that is mirror to a partial flag variety $\mathbb{ F}\ell(n_\bullet)$. Its Jacobi ring recovers the small quantum cohomology of $\mathbb{ F}\ell(n_\bullet)$ and we prove a…
We give an algorithm to compute the integer cohomology groups of any real partial flag manifold, by computing the incidence coefficients of the Schubert cells. For even flag manifolds we determine the integer cohomology groups, by proving…