Vanishing cotangent cohomology for Plücker algebras

We use representation theory and Bott’s theorem to show vanishing of higher cotangent cohomology modules for the homogeneous coordinate ring of Grassmannians in the Plücker embedding. As a by-product, we answer a question of Wahl about the cohomology of the square of the ideal sheaf for the case of Plücker relations. We obtain slightly weaker vanishing results for the cotangent cohomology of the coordinate rings of isotropic Grassmannians.     

线性子空间关系运算的Plücker坐标表示

本文讨论了线性子空间Plücker的线性表示,给出了Plücker关系式的矩阵形式,并由之导出了子空间关联关系、零化子空间、和子空间与交子空间Plücker坐标的矩阵表达式.     

On orthogonality-preserving plücker transformations of hyperbolic spaces

A complete overview of all orthogonality-preserving Plücker transformations in finite dimensional hyperbolic spaces with dimension other than three is given. In the Cayley-Klein model of such a hyperbolic space all Plücker transformations are induced by collineations of the ambient projective space.     

Toric degenerations of flag varieties from matching field tableaux

We present families of tableaux which interpolate between the classical semi-standard Young tableaux and matching field tableaux. Algebraically, this corresponds to SAGBI bases of Plücker algebras. We show that each such family of tableaux leads to a toric ideal, that can be realized as initial of the Plücker ideal, hence a toric degeneration for the flag variety.     

Differential geometry of grassmannians and the Plücker map

Using the Plücker map between grassmannians, we study basic aspects of classic grassmannian geometries. For ‘hyperbolic’ grassmannian geometries, we prove some facts (for instance, that the Plücker map is a minimal isometric embedding) that were previously known in the ‘elliptic’ case.     

Harmonic mappings and hyperbolic Plücker transformations

Let be a 3-dimensional hyperbolic space with Euclidean ground field . There is a certain procedure by which any harmonic mapping of the projective line over the unique quadratic extension of induces an orthogonality-preserving Plücker transformation of and, conversely, any orthogonality-preserving Plücker transformation of is induced by such a harmonic mapping. Received 30 July 1999; revised 25 November 1999.     

Symmetric functions,codes of partitions and the KP hierarchy

We consider an operator of Bernstein for symmetric functions and give an explicit formula for its action on an arbitrary Schur function. This formula is given in a remarkably simple form when written in terms of some notation based on the code of a partition. As an application, we give a new and very simple proof of a classical result for the KP hierarchy, which involves the Plücker relations for Schur function coefficients in a τ-function for the hierarchy. This proof is especially compact because we are able to restate the Plücker relations in a form that is symmetrical in terms of partition code notation.     

On resolving singularities of plane curves via a theorem attributed to Alfred Clebsch

This paper discusses a theorem in birational geometry that J.L. Coolidge attributed to Alfred Clebsch. The background is reconstructed from letters Felix Klein exchanged with Max Noether in 1894, when Noether was completing work on a lengthy report with Alexander Brill on the history of algebraic functions. Noether was deeply troubled to learn that Klein had informed him back in 1869 about relevant results that Clebsch and Leopold Kronecker had discussed in Berlin. These exchanges with Klein led to revisions in the Brill-Noether report, made in part to ensure Noether's own priority rights and larger intellectual legacy.     

A Primal-Dual Formulation for Certifiable Computations in Schubert Calculus

Formulating a Schubert problem as solutions to a system of equations in either Plücker space or local coordinates of a Schubert cell typically involves more equations than variables. We present a novel primal-dual formulation of any Schubert problem on a Grassmannian or flag manifold as a system of bilinear equations with the same number of equations as variables. This formulation enables numerical computations in the Schubert calculus to be certified using algorithms based on Smale’s \(\alpha \)-theory.     

Closed form of inverse Plücker correspondence in line geometry

Science China Mathematics - By mapping the homogeneous coordinates of two points in space to the Plücker coordinates of the line they determine, any transformation of type SL(4) upon points in...     

A note on Gröbner bases

In this paper we introduce the new notion of co-polynomials as polynomials arising from the Graßmann–Plücker polynomials. Pairs of co-polynomials are shown to be critical in the computation of a Gröbner basis for the chirotope ideal.     

Some necessary and sufficient conditions satisfied by decomposable tensors in grassmann spaces

The matrix form of the quadratic Plücker relations for decomposable tensors in Grassmann spaces is presented and some equivalent conditions are obtained. The proofs of the equivalence are elementary. The same methods are used to prove an analogous result for subspace incidence.     

On the Pontrjagin classes of a submanifold

In this paper the Pontrjagin forms of a closed oriented manifold immersed inR ⁿ are expressed in terms of Plücker coordinates. This work is supported partially by the National Natural Science Foundation of China.     

A Plücker formula for singular projective varieties

We prove a Plücker formula,for a projective variety X with arbitrary singularities, which expresses the class of X, the degree of the dual variety, in terms of Euler characteristics of X and of two linear sections of X. Moreover, we show that there is no formula whatsoever expressing this degree as a difference of two terms, a deformation invariant and a correction for singularities.     

Isometric embeddings of infinite-dimensional Grassmannians

We investigate geometric properties of the (Sato-Segal-Wilson) Grassmannian and its submanifolds, with special attention to the orbits of the KP flows. We use a coherentstates model, by which Spera and Wurzbacher gave equations for the image of a product of Grassmannians using the Powers-Størmer purification procedure. We extend to this product Sato’s idea of turning equations that define the projective embedding of a homogeneous space into a hierarchy of partial differential equations. We recover the BKP equations from the classical Segre embedding by specializing the equations to finite-dimensional submanifolds. We revisit the calculation of Calabi’s diastasis function given by Spera and Valli again in the context of C*-algebras, using the τ-function to give an expression of the diastasis on the infinitedimensional Grassmannian; this expression can be applied to the image of the Krichever map to give a proof of Weil’s reciprocity based on the fact that the distance of two points on the Grassmannian is symmetric. Another application is the fact that each (isometric) automorphism of the Grassmannian is induced by a projective transformation in the Plücker embedding.     

Poisson Brackets after Jacobi and Plücker

We construct a symplectic realization and a bi-Hamiltonian formulation of a 3-dimensional system whose solution are the Jacobi elliptic functions. We generalize this system and the related Poisson brackets to higher dimensions. These more general systems are parametrized by lines in projective space. For these rank 2 Poisson brackets the Jacobi identity is satisfied only when the Plücker relations hold. Two of these Poisson brackets are compatible if and only if the corresponding lines in projective space intersect. We present several examples of such systems.     

On reconstruction of a matrix by its minors

In this paper, we reconstruct matrices from their minors, and give explicit formulas for the reconstruction of matrices of orders 2 × 3, 2 × 4, 2 × n, 3 × 6 and m × n. We also formulate the Plücker relations, which are the conditions of the existence of a matrix related to its given minors.     

Discretization of asymptotic line parametrizations using hyperboloid surface patches

Two-dimensional affine A-nets in $3$ -space are quadrilateral meshes that discretize surfaces parametrized along asymptotic lines. The defining property of A-nets is planarity of vertex stars, hence elementary quadrilaterals of a generic A-net are skew. The present article deals with the extension of A-nets to differentiable surfaces, by gluing hyperboloid surface patches into the skew quadrilaterals. The obtained surfaces, named “hyperbolic nets”, are a novel, piecewise smooth discretization of surfaces parametrized along asymptotic lines. A simply connected affine A-net can be extended to a hyperbolic net if all quadrilateral strips are “equi-twisted”. The geometric condition of equi-twist implies the combinatorial property, that all inner vertices of the A-net have to be of even degree. If an A-net can be extended to a hyperbolic net, then there exists a 1-parameter family of such extensions. It is briefly explained how the generation of hyperbolic nets can be implemented on a computer. We use a projective model of Plücker line geometry in order to describe A-nets and hyperboloids.