An official website of the United States government
Positroids are certain representable matroids originally studied by Post- nikov in connection with the totally nonnegative Grassmannian and now used widely in algebraic combinatorics. The positroids give rise to determinantal equations defin- ing positroid varieties as subvarieties of the Grassmannian variety. Rietsch, Knutson– Lam–Speyer and Pawlowski studied geometric and cohomological properties of these varieties. In this paper, we continue the study of the geometric properties of positroid varieties by establishing several equivalent conditions characterizing smooth positroid varieties using a variation of pattern avoidance defined on decorated permutations, which are in bijection with positroids. Furthermore, we give a combinatorial method for determining the dimension of the tangent space of a positroid variety at key points using an induced subgraph of the Johnson graph. We also give a Bruhat interval char- acterization of positroids.
more »
« less
This content will become publicly available on April 15, 2027
On the parameters of line Schubert symplectic Grassmann codes
The Grassmannian [Formula: see text], is the collection of all two-dimensional subspaces of a vector space of dimension [Formula: see text]. It is one of the most widely studied objects in Algebraic Geometry and has interesting algebraic, geometric, and combinatorial properties. Since subspaces of dimension [Formula: see text] are known as lines the Grassmannian [Formula: see text], is known as the Grassmannian of lines. A class of linear codes, known as Grassmann codes, are used to understand the geometric and algebraic properties of the Grassmannian. Codes from Schubert subvarieties and polar subvarieties of the Grassmannian are known. For the case [Formula: see text], the parameters of Schubert codes have been established by Chen and the parameters of polar Grassmann codes have been established by Cardinali and Giuzzi. For general [Formula: see text], the parameters for Schubert codes are known, but the parameters of polar Grassmann codes are not known for general [Formula: see text]. The case in which either a polarity condition or a Schubert condition is applied to the Grassmannian on their own is simpler. These conditions have been studied independently but not concurrently. In this work, we study linear codes derived from subvarieties of the Grassmannian defined from the intersection of Schubert and symplectic varieties. We determine the length, dimension and minimum distance for several cases of line Schubert symplectic Grassmann codes when both the Schubert conditions and some special symplectic polarity conditions are combined in different ways.
more »
« less
- Award ID(s):
- 2150434
- PAR ID:
- 10688517
- Publisher / Repository:
- Journal of Algebra and its Applications
- Date Published:
- Journal Name:
- Journal of Algebra and Its Applications
- ISSN:
- 0219-4988
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
In this paper, we construct codes with hierarchical locality using natural geometric structures in Artin–Schreier surfaces of the form [Formula: see text]. Our main theorem describes the codes, their hierarchical structure and recovery algorithms, and gives parameters. We also develop a family of examples using codes defined over [Formula: see text] on the surface [Formula: see text]. We use elementary methods to count the [Formula: see text]-rational points on the surface, enabling us to provide explicit hierarchical parameters and a better bound on minimum distance for these codes. An additional example and some generalizations are also considered.more » « less
-
[Formula: see text]A hyperbolic code is an evaluation code that improves a Reed–Muller code because the dimension increases while the minimum distance is not penalized. We give necessary and sufficient conditions, based on the basic parameters of the Reed–Muller code, to determine whether a Reed–Muller code coincides with a hyperbolic code. Given a hyperbolic code [Formula: see text], we find the largest Reed–Muller code contained in [Formula: see text] and the smallest Reed–Muller code containing [Formula: see text]. We then prove that similar to Reed–Muller and affine Cartesian codes, the [Formula: see text]th generalized Hamming weight and the [Formula: see text]th footprint of the hyperbolic code coincide. Unlike for Reed–Muller and affine Cartesian codes, determining the [Formula: see text]th footprint of a hyperbolic code is still an open problem. We give upper and lower bounds for the [Formula: see text]th footprint of a hyperbolic code that, sometimes, are sharp.more » « less
-
We consider the topological and geometric reconstruction of a geodesic subspace of [Formula: see text] both from the Čech and Vietoris-Rips filtrations on a finite, Hausdorff-close, Euclidean sample. Our reconstruction technique leverages the intrinsic length metric induced by the geodesics on the subspace. We consider the distortion and convexity radius as our sampling parameters for the reconstruction problem. For a geodesic subspace with finite distortion and positive convexity radius, we guarantee a correct computation of its homotopy and homology groups from the sample. This technique provides alternative sampling conditions to the existing and commonly used conditions based on weak feature size and [Formula: see text]–reach, and performs better under certain types of perturbations of the geodesic subspace. For geodesic subspaces of [Formula: see text], we also devise an algorithm to output a homotopy equivalent geometric complex that has a very small Hausdorff distance to the unknown underlying space.more » « less
-
We extend the self-organizing mapping algorithm to the problem of visualizing data on Grassmann manifolds. In this setting, a collection of k points in n-dimensions is represented by a k-dimensional subspace, e.g., via the singular value or QR-decompositions. Data assembled in this way is challenging to visualize given abstract points on the Grassmannian do not reside in Euclidean space. The extension of the SOM algorithm to this geometric setting only requires that distances between two points can be measured and that any given point can be moved towards a presented pattern. The similarity between two points on the Grassmannian is measured in terms of the principal angles between subspaces, e.g., the chordal distance. Further, we employ a formula for moving one subspace towards another along the shortest path, i.e., the geodesic between two points on the Grassmannian. This enables a faithful implementation of the SOM approach for visualizing data consisting of k-dimensional subspaces of n-dimensional Euclidean space. We illustrate the resulting algorithm on a hyperspectral imaging application.more » « less
