Plucker Coordinates Research Articles

Edge vectors on plabic networks in the disk and amalgamation of totally non-negative Grassmannians

The amalgamation of cluster varieties introduced by Fock and Goncharov in [21] plays a relevant role both in mathematical and physical problems. In particular, amalgamation in the totally non-negative part of positroid varieties is explicitly described as gluing of several copies of small positive Grassmannians, GrTP(1,3) and GrTP(2,3), has important topological implications [53] and naturally appears in the computation of amplitude scatterings in N=4 SYM theory [9,10]. Lam [44] has proposed to represent amalgamation in positroid varieties by equivalence classes of relations on bipartite graphs and to identify total non-negativity via appropriate edge signatures. In this paper we provide an explicit geometric characterization of such signatures in the setting of the planar bicolored trivalent directed perfect networks in the disk introduced in [52] to parametrize positroid cells SMTNN⊂GrTNN(k,n) using systems of relations for n-vectors.More precisely, to any such graph G, we associate a geometric signature satisfying both the full rank condition and the total non–negativity property on the full positroid cell. Such signature is uniquely identified by geometric indices (local winding and intersection number) ruled by the orientation O and gauge ray direction l on G.The principal result is the enrichment of Postnikov's construction in [52] by associating measurements not only to boundary edges or vertices, but to internal edges as well. Indeed we generalize prior results by Postnikov [52] and Talaska [61] providing an explicit representation of the solution to the system of geometric relations on the network (N,O,l) of graph G and positive weights. At this aim, we assign canonical basis vectors in Rn at the boundary sinks and define the vectors components at the edge e as (finite or infinite) summations over the directed paths from e to the given boundary sink. Such edge vectors have the following properties:1) They solve the geometric system of relations on (N,O,l);2) Their components are rational in the weights with subtraction–free denominators, and have explicit expressions in terms of the conservative and edge flows of [61]. At the boundary sources they coincide with the entries of the boundary measurement matrix defined in [52]. If N is acyclically orientable, all components are subtraction–free rational expressions in the weights with respect to a convenient basis. Null edge vectors may occur on reducible networks not acyclically orientable;3) We provide explicit formulas both for the transformation rules of the edge vectors with respect to the orientation and the several gauges of the given network, and for their transformations due to moves and reductions of networks.Finally, we associate a Kasteleyn orientation to the graph following [17]. If the graph is bipartite, it is known that the partition functions of dimer configurations on the graph with given boundary conditions coincide with the Plucker coordinates of the corresponding point of the totally non-negative Grassmannian [54,44,60,1,8]. In the case of plabic graphs which are not bipartite we show that the partition function for a given boundary condition is not a multiple of the corresponding minor of the boundary measurement matrix. Therefore, in this case a statistical mechanical interpretation of the boundary measurement map remains open.

A SLAM System Based on RGBD Image and Point-Line Feature

Most of the existing visual SLAM schemes rely solely on point or line features to estimate camera trajectory. In some scenes such as texture missing or motion blurring, it’s difficult to find a sufficient number of reliable features, resulting in low positioning accuracy. To extract more features, an RPL-SLAM solution is proposed to extract point features and line features respectively. Ulteriorly, the depth information of RGBD image is used to restore the 3D information of point and line features, improving the accuracy of camera track positioning. RPL-SLAM scheme mainly includes three modules: tracking, local mapping and loop detection. Tracking module extends the application of line feature on the basis of point feature extraction and matching. A SLD line segment extraction algorithm which can eliminate the micro segments and a DBM segment matching algorithm based on word bag are proposed respectively. These two algorithms improve the matching efficiency while ensuring the matching accuracy, and effectively track and locate the camera of each frame. In the local mapping and loop detection module, the Plucker coordinates are applied to express the spatial line and define the re-projection error of the straight line, so that the back-end optimization error model of point-line fusion is unified to solve the instability problem in optimization. RPL-SLAM is tested on TUM RGBD and ICL-NUIM data set respectively, and compared with ORB-SLAM2. The result shows that RPL-SLAM can effectively improve the accuracy of pose estimation and map reconstruction while maintaining real-time performance by fusing point-line features with depth images.

Definable relations in finite dimensional subspace lattices with involution. Part II: Quantifier-free and homogeneous descriptions

For finite dimensional hermitean inner product spaces V, over \(*\)-fields F, and in the presence of orthogonal bases providing form elements in the prime subfield of F, we show that quantifier-free definable relations in the subspace lattice \(\mathsf{L}(V)\), endowed with the involution induced by orthogonality, admit quantifier-free descriptions within F, also in terms of Grassmann–Plucker coordinates. In the latter setting, homogeneous descriptions are obtained if one allows quantification type \(\Sigma _1\). In absence of involution, these results remain valid.

Closed form of inverse Plücker correspondence in line geometry

By mapping the homogeneous coordinates of two points in space to the Plucker coordinates of theline they determine, any transformation of type $SL(4)$ upon points in space is mapped to a transformationof type $SO_0(3,3)$, the latter being the connected componentcontaining the identity of the special orthogonal transformation group of the linear spacespanned by Plucker coordinates. This is the classical Plucker correspondence, two-to-one and onto. It has importantapplications in line geometry and projective transformations. While the explicit form of Plucker correspondence is trivialto present, its inverse in explicit form, which is also important in application, is not found in the literature.In this paper, we present a simple and unified formula for the inverse of the Plucker correspondence.

Robust Cochlear Modiolar Axis Detection in CT.

The cochlea, the auditory part of the inner ear, is a spiral-shaped organ with large morphological variability. An individualized assessment of its shape is essential for clinical applications related to tonotopy and cochlear implantation. To unambiguously reference morphological parameters, reliable recognition of the cochlear modiolar axis in computed tomography (CT) images is required. The conventional method introduces measurement uncertainties, as it is based on manually selected and difficult to identify landmarks. Herein, we present an algorithm for robust modiolar axis detection in clinical CT images. We define the modiolar axis as the rotation component of the kinematic spiral motion inherent in the cochlear shape. For surface fitting, we use a compact shape representation in a 7-dimensional kinematic parameter space based on extended Plücker coordinates. It is the first time such a kinematic representation is used for shape analysis in medical images. Robust surface fitting is achieved with an adapted approximate maximum likelihood method assuming a Student-t distribution, enabling axis detection even in partially available surface data. We verify the algorithm performance on a synthetic data set with cochlear surface subsets. In addition, we perform an experimental study with four experts in 23 human cochlea CT data sets to compare the automated detection with the manually found axes. Axes found from co-registered high resolution μCT scans are used for reference. Our experiments show that the algorithm reduces the alignment error providing more reliable modiolar axis detection for clinical and research applications.

Screw Theory in Lorentzian Space

In this paper we present various results about the six dimensional vectors obtained from the tangent operators of spatial motion, called as screws, in Lorentzian space. Each screw has an axis defined by six Plucker coordinates in Lorentzian space. The manipulation of screw coordinate transformations has been simplified by using Lorentz matrix multiplication and dual number algebra. Also, we showed that screw displacement is representation as the exponential of a dual angular velocity matrix by using the dual orthogonal matrices in Lorentzian space.

Pure Condition of Both 6R Serial and 3-PRS Parallel Robots Using Grassmann-Cayley Algebra

This research presents Pure Condition approach, which has used in analyzing simultaneously the singularity configuration and the rigidity of mechanism. The study cases analysis is implemented on variable joints orientation of 6R (Revolute) Serial Manipulators (SMs) and variable actuated joints position of 3-PRS (Prismatic-Revolute-Spherical) Parallel Manipulators (PMs) using Grassmann-Cayley Algebra (GCA). In this work we require in Projective Space both Twist System (TS) and Global Wrench System (GWS) respectively for serial and parallel manipulators which represent the Jacobian Matrix (J) in symbolic approach to Plucker coordinate vector of lines and unify framework on static and kinematics respectively. This paper, works, is designed to determine geometrically at symbolic level the vanished points of inverse form of this Jacobian Matrix (J) which called superbracket in GCA. The investigation vary to those reported early by introducing GCA approach on the singularity of serial robot, variable joints orientation and actuated positions on robot manipulators (RMs) to analyze rigidity frame work and singularity configuration which involve simultaneously Pure Condition. And the results also revealed a single singularity condition which contains all particulars cases and three general cases such as the shoulder, elbow and wrist singularity for SMs while double, single and undermined singularities respectively for 3-PRS, 3-PRS and 3-PRS PMs which contain all generals and particulars cases.

On the arithmetic 4-orbifolds associated to integral quaternary quadratic forms of index 2

Groups acting properly and discontinuously on the Cartesian product $$\mathbb {H}^{2}\times \mathbb {H}^{2}$$ of two hyperbolic planes are termed hyperabelian by Picard. The automorphism group $$\mathrm {Aut}f$$ of a quaternary integral quadratic form f of index 2 is an example of a hyperabelian group. Hence the quotient orbifold $$Q_{f}$$ of the action of $$\mathrm {Aut}f$$ on $$\mathbb {H}^{2}\times \mathbb {H}^{2}$$ is a 4-dimensional arithmetic orbifold, endowed with a natural $$\mathbb {H}^{2}\times \mathbb {H}^{2}$$ -geometry. Plucker coordinates are used to understand $$Q_{f}$$ . A real automorphism U of $$\mathbb {R}^{4}$$ induces a real automorphism $$\mathbf {K(}U)$$ of $$(\mathbb {R}^{6},k)$$ in such a way that if $$U\in SL(4,\mathbb {Z})$$ then $$\mathbf {K(}U)\in SL(6,\mathbb {Z})$$ is an automorph of the Klein quadratic form k. It is proved that the converse is true. That is, given an automorph $$M\in SL(6,\mathbb {Z})$$ of k there is $$U\in SL(4,\mathbb {Z})$$ such that $$\mathbf {K(}U)=\pm M$$ , so that the proper automorphism group of the Klein quadric is isomorphic to $$SL(4,\mathbb {Z})$$ via $$\mathbf {K}$$ . This is used to obtain the automorphism group of the quadratic line complex of line tangents to a quadric in projective space $$P^{3}$$ . With this, a description is given of the automorphism group of a quaternary integral quadratic form of index 2.

3-D Reconstruction of Binocular Vision Using Distance Objective Generated From Two Pairs of Skew Projection Lines

A reconstruction method is provided to improve the measurement of the binocular vision. The image projections of the 3-D point on the 3-D reference are analyzed by the multiple view geometry. The Plucker coordinates of the two screw projection lines are constructed by the image points. The line segment perpendicular to two projection lines is generated from the direction vector and the end points of the line segment. As the bilinear products of the Plucker coordinates between the perpendicular line and the projection lines are zero, the end points of the perpendicular line segment are parameterized by the Cramer’s Rule. The reconstruction error of the distance between two parameterized 3-D points is employed as the minimized objective of the cost function. The impact factors and the enhancements of the reconstruction accuracy are investigated by the experiments comparing with the original method. The method contributes the error descents of 27.19%, 38.29%, 19.01%,and 16.31% for the test lengths of 50, 100, 150, and 200 mm. Therefore, the results demonstrate that reconstruction method of the binocular vision provides the higher accuracy and application potentials.

Matrices of SL $${(4, \mathbb{R})}$$ ( 4 , R ) that are the Product of Two Skew-Symmetric Matrices

The Jordan forms of matrices that are the product of two skew-symmetric matrices over a field of characteristic \({\neq 2}\) have been a research topic in linear algebra since the early twentieth century. For such a matrix, its Jordan form is not necessarily real, nor does the matrix similarity transformation change the matrix into the Jordan form. In 3-D oriented projective geometry, orientation-preserving projective transformations are matrices of \({SL(4, \mathbb{R})}\), and those matrices of \({SL(4, \mathbb{R})}\) that are the product of two skew-symmetric matrices are the generators of the group \({SL(4, \mathbb{R})}\). The canonical forms of orientation-preserving projective transformations under the group action of \({SL(4, \mathbb{R})}\)-similarity transformations, called \({SL(4, \mathbb{R})}\)-Jordan forms, are more useful in geometric applications than complex-valued Jordan forms. In this paper, we find all the \({SL(4, \mathbb{R})}\)-Jordan forms of the matrices of \({SL(4, \mathbb{R})}\) that are the product of two skew-symmetric matrices, and divide them into six classes, so that each class has an unambiguous geometric interpretation in 3-D oriented projective geometry. We then consider the lifts of these transformations to SO(3, 3) by extending the action of \({SL(4, \mathbb{R})}\) from points to lines in space, so that in the vector space \({\mathbb{R}^{3, 3}}\) spanned by the Plucker coordinates of lines these projective transformations become special orthogonal transformations, and the six classes are lifted to six different rotations in 2-D planes of \({\mathbb{R}^{3, 3}}\).

Circle Complexes and the Discrete CKP Equation

In the spirit of Klein's Erlangen Program, we investigate the geometric and algebraic structure of fundamental line complexes and the underlying privileged discrete integrable system for the minors of a matrix which constitute associated Plucker coordinates. Particular emphasis is put on the restriction to Lie circle geometry which is intimately related to the master dCKP equation of discrete integrable systems theory. The geometric interpretation, construction and integrability of fundamental line complexes in Mobius, Laguerre and hyperbolic geometry are discussed in detail. In the process, we encounter various avatars of classical and novel incidence theorems and associated cross- and multi-ratio identities for particular hypercomplex numbers. This leads to a discrete integrable equation which, in the context of Mobius geometry, governs novel doubly hexagonal circle patterns.

Extensors and the Hilbert scheme

The Hilbert scheme $Hilb_{p(t)}^n$ parametrizes closed subschemes and families of closed subschemes in the projective space $P^n$ with a fixed Hilbert polynomial $p(t)$. It is classically realized as a closed subscheme of a Grassmannian or a product of Grassmannians. In this paper we present a method that allows to derive scheme theoretical global equations for $Hilb_{p(t)}^n$ in the Plucker coordinates of a Grassmannian $Grass_p^N$, where $p$ and $N$ depend on the dimension $n$ of the projective space and on the Hilbert polynomial $p(t)$. Using this method we obtain the already known set of equations given by Iarrobino and Kleiman in 1999, the one conjectured by Bayer in 1982 and proved by Haiman and Sturmfels in 2004, and also a new set of equations of degree lower than the previous ones. The novelties of our approach are essentially two. The first one is a local study of the Hilbert functor through special sets of open subfunctors obtained exploiting the symmetries of the Hilbert scheme and the combinatorial properties of monomial ideals, mainly the Borel-fixed ones. The second one is a generalization of the theory of extensors to the setting of free modules over any ring $A$ and the description of any exterior product of elements of a free submodule in terms of Plucker coordinates.

Twists of Plücker Coordinates as Dimer Partition Functions

The homogeneous coordinate ring of the Grassmannian Gr k,n has a cluster structure defined in terms of planar diagrams known as Postnikov diagrams. The cluster corresponding to such a diagram consists entirely of Plucker coordinates. We introduce a twist map on Gr k,n , related to the Berenstein–Fomin–Zelevinsky-twist, and give an explicit Laurent expansion for the twist of an arbitrary Plucker coordinate in terms of the cluster variables associated with a fixed Postnikov diagram. The expansion arises as a (scaled) dimer partition function of a weighted version of the bipartite graph dual to the Postnikov diagram, modified by a boundary condition determined by the Plucker coordinate. We also relate the twist map to a maximal green sequence.

New method for constructing semi-invariants and integrals of the full symmetric $\mathfrak{s}\mathfrak{l}_n $ Toda lattice

We consider the full symmetric representation of the Lax operator matrix of the Toda lattice, which is known as the full symmetric Toda lattice. The phase space of this system is the generic orbit of the coadjoint action of the Borel subgroup B n + of SL n (ℝ). This system is integrable. We propose a new method for constructing semi-invariants and integrals of the full symmetric Toda lattice. Using only the equations of motion for the Lax eigenvector matrix, we prove the existence of the semi-invariants that are Plucker coordinates in the corresponding projective spaces. We use these semi-invariants to construct the integrals. Our new approach provides simple exact formulas for the full set of independent semi-invariants and integrals expressed in terms of the Lax matrix and also in terms of its eigenvector and eigenvalue matrices of the full symmetric Toda lattice without using the chopping and Kostant procedures. We describe the structure of the additional integrals of motion as functions on the flag space modulo the Toda flows and show how the Plucker coordinates of different projective spaces define different families of the additional integrals.

Constructing the quantum Hall system on the Grassmannians Gr2(CN)

In this talk, we give the formulation of Quantum Hall Effects (QHEs) on the complex Grassmann manifolds Gr2(CN). We set up the Landau problem in Gr2(CN), solve it using group theoretical techniques and provide the energy spectrum and the eigenstates in terms of the SU(N) Wigner D-functions for charged particles on Gr2(CN) under the influence of abelian and non-abelian background magnetic monopoles or a combination of these thereof. For the simplest case of Gr2(C4) we provide explicit constructions of the single and many- particle wavefunctions by introducing the Plucker coordinates and show by calculating the two-point correlation function that the lowest Landau level (LLL) at filling factor v = 1 forms an incompressible fluid. Finally, we heuristically identify a relation between the U(1) Hall effect on Gr2(C4) and the Hall effect on the odd sphere S5, which is yet to be investigated in detail, by appealing to the already known analogous relations between the Hall effects on CP3 and CP7 and those on the spheres S4 and S8, respectively. The talk is given by S. Kürkçüoğlu at the Group 30 meeting at Ghent University, Ghent, Belgium in July 2014 and based on the article by F.Ballı, A.Behtash, S. Kürkçüoğlu, G.Ünal [1].

Sign variation, the Grassmannian, and total positivity

The <i>totally nonnegative Grassmannian</i> is the set of $k$-dimensional subspaces $V$ of ℝ<sup>$n$</sup> whose nonzero Plücker coordinates (i.e. $k × k$ minors of a $k × n$ matrix whose rows span $V$) all have the same sign. Total positivity has been much studied in the past two decades from an algebraic, combinatorial, and topological perspective, but first arose in the theory of oscillations in analysis. It was in the latter context that Gantmakher and Krein (1950) and Schoenberg and Whitney (1951) independently showed that a subspace $V$ is totally nonnegative iff every vector in $V$, when viewed as a sequence of $n$ numbers and ignoring any zeros, changes sign fewer than $k$ times. We generalize this result, showing that the vectors in $V$ change sign fewer than $l$ times iff certain sequences of the Plücker coordinates of some <i>generic perturbation</i> of $V$ change sign fewer than $l − k + 1$ times. We give an algorithm which constructs such a generic perturbation. Also, we determine the <i>positroid cell</i> of each totally nonnegative $V$ from sign patterns of vectors in $V$. These results generalize to oriented matroids. La <i>grassmannienne totalement non négative</i> est l’ensemble des sous-espaces $V$ de ℝ<sup>$n$</sup> de dimension $k$ dont coordonnées plückeriennes non nulles (mineurs de l’ordre $k$ d’une matrice $k × n$ dont les lignes engendrent $V$) ont toutes le même signe. La positivité totale a beaucoup été étudiée durant les deux dernières décennies d’une perspective algébrique, combinatoire, et topologique, mais a pris naissance dans la théorie analytique des oscillations. C’est dans ce contexte que Gantmakher et Krein (1950) et Schoenberg et Whitney (1951) ont indépendamment démontré qu’un sous-espace $V$ est totalement non négatif ssi chaque vecteur dans $V$, lorsque considéré comme une séquence de $n$ nombres et dont on ignore les zéros, change de signe moins de $k$ fois. Nous généralisons ce résultat, démontrant que les vecteurs dans $V$ changent de signe moins de $l$ fois ssi certaines séquences des coordonnées plückeriennes d’une <i>perturbation générique</i> de $V$ changent de signe moins de $l − k + 1$ fois. Un algorithme construisant une telle perturbation générique est obtenu. De plus, nous déterminons la <i>cellule positroïde</i> de chaque $V$ totalement non négatif à partir des données de signe des vecteurs dans $V$. Ces résultats sont valides pour les matroïdes orientés.

Network parametrizations for the Grassmannian

Deodhar introduced his decomposition of partial flag varieties as a tool for understanding Kazhdan-Lusztig polynomials. The Deodhar decomposition of the Grassmannian is also useful in the context of soliton solutions to the KP equation, as shown by Kodama and the second author. Deodhar components S_D of the Grassmannian are in bijection with certain tableaux D called Go-diagrams, and each component is isomorphic to (K*)^a \times (K)^b for some non-negative integers a and b. Our main result is an explicit parameterization of each Deodhar component in the Grassmannian in terms of networks. More specifically, from a Go-diagram D we construct a weighted network N_D and its weight matrix W_D, whose entries enumerate directed paths in N_D. By letting the weights in the network vary over K or K* as appropriate, one gets a parameterization of the Deodhar component S_D. One application of such a parameterization is that one may immediately determine which Plucker coordinates are vanishing and nonvanishing, by using the Lindstrom-Gessel-Viennot Lemma. We also give a (minimal) characterization of each Deodhar component in terms of Plucker coordinates. A main tool for us is the work of Marsh and Rietsch on Deodhar components in the flag variety.

A Formula for Plucker Coordinates Associated with a Planar Network

For a planar directed graph G, Postnikov's boundary measurement map sends positive weight functions on the edges of G onto the appropriate totally non-negative Grassmann cell. We establish an explicit formula for Postnikov's map by expressing each Plucker coordinate as a ratio of two combinatorially defined polynomials in the edge weights, with positive integer coefficients. In the non-planar setting, we show that a similar formula holds for special choices of Plucker coordinates.

On surfaces immersed in Euclidean space R 4

Let M be a closed oriented surface immersed in R4. Associated it one has the generalized Gauss map from M into the Grassmann manifold G4,2. This note will be concerned with the geometry of the generalized Gauss map by using the moving frame theory and the quaternion interpretation of Plucker coordinates. As one of consequences, we get the celebrated theorem of Chern and Spanier, Hoffman and Osserman, who proved it by quite different methods. At last, we give an explicit construction of a series of immersions of S2 in R4 with any given normal Euler number.

Quantization of flag manifolds and their supersymmetric extensions

We first review the description of flag manifolds in terms of Plucker coordinates and coherent states. Using this description, we construct fuzzy versions of the algebra of functions on these spaces in both operatorial and star product language. Our main focus is here on flag manifolds appearing in the double fibration underlying the most common twistor correspondences. After extending the Plucker description to certain supersymmetric cases, we also obtain the appropriate deformed algebra of functions on a number of fuzzy flag supermanifolds. In particular, fuzzy versions of Calabi–Yau supermanifolds are found.