Some partial formulae for Stiefel-Whitney classes of Grassmannians (original) (raw)

On the dual Stiefel-Whitney classes of some Grassmann manifolds

Acta Mathematica Hungarica, 2009

We present some non-vanishing dual Stiefel-Whitney classes of the Grassmann manifolds O(n)/O(4) × O(n − 4) for n = 2s + 2 and n = 2s + 3 (s ≧ 3), providing a supplement to results of Hiller, Stong, and Oproiu. Some applications are also mentioned.

A dual approach to structure constants for K-theory of Grassmannians

Discrete Mathematics & Theoretical Computer Science

The problem of computing products of Schubert classes in the cohomology ring can be formulated as theproblem of expanding skew Schur polynomial into the basis of ordinary Schur polynomials. We reformulate theproblem of computing the structure constants of the Grothendieck ring of a Grassmannian variety with respect to itsbasis of Schubert structure sheaves in a similar way; we address the problem of expanding the generating functions forskew reverse-plane partitions into the basis of polynomials which are Hall-dual to stable Grothendieck polynomials. From this point of view, we produce a chain of bijections leading to Buch’s K-theoretic Littlewood-Richardson rule.

A G ] 7 A ug 2 01 5 Degree Formulae for Grassmann Bundles , II

2018

Abstract. Let X be a non-singular quasi-projective variety over a field, and let E be a vector bundle over X . Let GX(d, E) be the Grassmann bundle of E overX parametrizing corank d subbundles of E with projection π : GX(d, E) → X , and let Q ← π ∗E be the universal quotient bundle of rank d. In this article, a closed formula for π∗ ch(detQ), the push-forward of the Chern character of the Plücker line bundle detQ by π is given in terms of the Segre classes of E . Our formula yields a degree formula for GX(d, E) with respect to detQ when X is projective and ∧E is very ample. To prove the formula above, a push-forward formula in the Chow rings from a partial flag bundle of E to X is given.

Three lectures on quiver Grassmannians

arXiv: Algebraic Geometry, 2020

This paper contains the material discussed in the series of three lectures that I gave during the workshop of the ICRA 2018 in Prague. I will introduce the reader to some of the techniques used in the study of the geometry of quiver Grassmannians. The notes are quite elementary and thought for phd students or young researchers. I assume that the reader is familiar with the representation theory of quivers.

The generating rank of the unitary and symplectic Grassmannians

Journal of Combinatorial Theory, Series A, 2012

We prove that the Grassmannian of totally isotropic k-spaces of the polar space associated to the unitary group SU 2n (F) (n ∈ N) has generating rank 2n k when F = F 4 . We also reprove the main result of Blok [4], namely that the Grassmannian of totally isotropic k-spaces associated to the symplectic group Sp 2n (F) has generating rank 2n k − 2n k−2 , when Char(F) = 2.

On the classification of homogeneous 222-spheres in complex Grassmannians

Osaka Journal of Mathematics, 2013

In this paper we discuss a classification problem of homogene ous 2-spheres in the complex Grassmann manifold G(kC1,nC1) by theory of unitary representations of the 3-dimensional special unitary group SU(2). First we observe that if an immersion x W S2 ! G(k C 1, nC 1) is homogeneous, then its image x(S2) is a 2-dimensional (SU(2))-orbit in G(kC1,nC1), where W SU(2)! U (nC1) is a unitary representation of SU(2). Then we give a classification theorem of homogeneous 2-s pheres in G(kC1,nC1). As an application we describe explicitly all homogeneou s 2-spheres in G(2, 4). Also we mention about an example of non-homogeneous h lomorphic 2-sphere with constant curvature in G(2, 4).

Combinatorial models for the finite-dimensional Grassmannians

Discrete & Computational Geometry, 1993

Let ~t 'n be a linear hyperplane arrangement in R n. We define two corresponding posets ff~(~r and ~k(Jt'*) of oriented matroids, which approximate the Grassmannian Gk(R ~) and the Stiefel manifold Vk(R"). The basic conjectures are that the "OM-Grassmannian" f~k(Jr has the homotopy type of Gk(~n), and that the "OM-Stiefel bundle" An: A~k(J[ ~)-* Af~k(Jt '") is a surjective map. These conjectures can be proved in some cases: we survey the known results and add some new ones. The conjectures fail if they are generalized to nonrealizable oriented matroids .gcn.

The double Cayley Grassmannian

arXiv: Algebraic Geometry, 2020

We study the smooth projective symmetric variety of Picard number one that compactifies the exceptional complex Lie group G2, by describing it in terms of vector bundles on the spinor variety of Spin(14). We call it the double Cayley Grassmannian because quite remarkably, it exhibits very similar properties to those of the Cayley Grassmannian (the other symmetric variety of type G2), but doubled in the certain sense. We deduce among other things that all smooth projective symmetric varieties of Picard number one are infinitesimally rigid.

Equivariant Giambelli and Determinantal Restriction Formulas for the Grassmannian

Pure and Applied Mathematics Quarterly, 2006

The main result of the paper is a determinantal formula for the restriction to a torus fixed point of the equivariant class of a Schubert subvariety in the torus equivariant integral cohomology ring of the Grassmannian. As a corollary, we obtain an equivariant version of the Giambelli formula.