Related papers: Boolean Schubert Structure Coefficients

Let g be a Lie algebra of type A,D,E with fixed Cartan subalgebra h, root system R and Weyl group W. We show that a choice of Coxeter element C gives a root basis for g. Moreover we show that this root basis gives a purely combinatorial…

We give the formula for multiplying a Schubert class on an odd orthogonal or symplectic flag manifold by a special Schubert class pulled back from a Grassmannian of maximal isotropic subspaces. This is also the formula for multiplying a…

The quantum cohomology algebra of the (full) flag manifold is a fundamental example in quantum cohomology theory, with connections to combinatorics, algebraic geometry, and integrable systems. Using a differential geometric approach, we…

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…

In Schubert Puzzles and Integrability I we proved several "puzzle rules" for computing products of Schubert classes in K-theory (and sometimes equivariant K-theory) of d-step flag varieties. The principal tool was "quantum integrability",…

We determine explicitly the irreducible components of the singular locus of any Schubert variety for GL_n(K), K being an algebraically closed field of arbitrary characteristic. We also describe the generic singularities along these…

We propose a combinatorial model for the Schubert structure constants of the complete flag manifold when one of the factors is Grassmannian.

Stable cohomology is a generalization of Tate cohomology to associative rings, first defined by Pierre Vogel. For a commutative local ring $R$ with residue field $k$, stable cohomology modules $\widehat{\mathrm{Ext}}{\vphantom…

The ring K(G/B) is isomorphic to a quotient of a Laurent polynomial ring by an ideal generated by certain W-symmetric functions and has a basis given by classes O_w, where O_w is the class of the structure sheaf of the Schubert variety X_w.…

This article deals with the quotient category of the category of coherent sheaves on an irreducible smooth projective variety by the full subcategory of sheaves supported in codimension greater than c. It turns out that this category has…

The problem of gauging a closed form is considered. When the target manifold is a simple Lie group G, it is seen that there is no obstruction to the gauging of a subgroup H\subset G if we may construct from the form a cocycle for the…

In this paper, we present a closed formula for the cohomology of real Grassmannians. To achieve this, we use a theory of stratified spaces to compute the differentials in a chain complex that computes the cohomology. Specifically, we…

We obtain a class of locally symetric Kaehler Einstein structures on the nonzero cotangent bundle of a Riemannian manifold of positive constant sectional curvature. The obtained class of Kaehler Einstein structures depends on one essential…

For each infinite series of the classical Lie groups of type B,C or D, we introduce a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in…

Schubert polynomials were introduced in the context of the geometry of flag varieties. This paper investigates some of the connections not yet understood between several combinatorial structures for the construction of Schubert polynomials;…

We obtain a class of Kaehler Einstein structures on the nonzero cotangent bundle of a Riemannian manifold of positive constant sectional curvature. The obtained class of Kaehler Einstein structure depends on one essential parameter, cannot…

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…

Billey and Braden defined maps on flag manifolds that are the geometric counterpart of permutation patterns. A section of their pattern map is an embedding of the flag manifold of a Levi subgroup into the full flag manifold. We give two…

We introduce an additive basis of the integral cohomology ring of the Peterson variety which reflects the geometry of certain subvarieties of the Peterson variety. We explain the positivity of the structure constants from a geometric…

We investigate Lie bialgebra structures on simple Lie algebras of non-split type $A$. It turns out that there are several classes of such Lie bialgebra structures, and it is possible to classify some of them. The classification is obtained…