Related papers: Enumerative Coding for Line Polar Grassmannians wi…

This book is dedicated to Grassmannians associated with buildings of classical types: usual, polar, and half-spin Grassmannians. Grassmannians of vector spaces and Grassmannians consisting of totally isotropic subspaces of non-degenerate…

Sparsity-based representations have recently led to notable results in various visual recognition tasks. In a separate line of research, Riemannian manifolds have been shown useful for dealing with features and models that do not lie in…

Polar Grassmann codes of orthogonal type have been introduced in I. Cardinali and L. Giuzzi, \emph{Codes and caps from orthogonal Grassmannians}, {Finite Fields Appl.} {\bf 24} (2013), 148-169. They are subcodes of the Grassmann code…

In this manuscript, we introduce a new class of linear codes, called affine symplectic Grassmann codes, and determine their parameters, automorphism group, minimum distance codewords, dual code and other key features. These linear codes are…

For an arbitrary field of any characteristic we give an explicit description, in terms of Pl\"ucker coordinates, of the projective linear space that cuts out the Lagrangian-Grassmannian variety $L(n,2n)$ of maximal isotropic subspaces in a…

Let $\widetilde{I}_{2n,k}$ denote the space of $k$-dimensional, oriented isotropic subspaces of $\mathbb{R}^{2n}$, called the oriented isotropic Grassmannian. Let $f \colon \widetilde{I}_{2n,k} \rightarrow \widetilde{I}_{2m,l} $ be a map…

The regular point-line geometry with respect to a pseudo-polarity is introduced. It is weaker than the underlying metric-projective geometry. The automorphism group of this geometry is determined. This geometry can be also expressed as the…

Odd symplectic Grassmannians are a generalization of symplectic Grassmannians to odd-dimensional spaces. Here we compute the classical and quantum cohomology of the odd symplectic Grassmannian of lines. Although these varieties are non…

The Grassmannian of affine subspaces is a natural generalization of both the Euclidean space, points being zero-dimensional affine subspaces, and the usual Grassmannian, linear subspaces being special cases of affine subspaces. We show…

In this paper we relate a well-known in symplectic geometry compactification of the space of symmetric bilinear forms considered as a chart of the Lagrangian Grassmannian to the specific compactifications of the space of electrical networks…

Let $V$ and $V'$ be vector spaces of dimension $n$ and $n'$, respectively. Let $k\in\{2,...,n-2\}$ and $k'\in\{2,...,n'-2\}$. We describe all isometric and $l$-rigid isometric embeddings of the Grassmann graph $\Gamma_{k}(V)$ in the…

Let $\Gamma(n,k)$ be the Grassmann graph formed by the $k$-dimensional subspaces of a vector space of dimension $n$ over a field $\mathbb F$ and, for $t\in \mathbb{N}\setminus \{0\}$, let $\Delta_t(n,k)$ be the subgraph of $\Gamma(n,k)$…

We first prove an identity involving symmetric polynomials. This identity leads us into exploring the geometry of Lagrangian Grassmannians. As an insight applications, we obtain a formula for the integral over the Lagrangian Grassmannian of…

Let $V$ be an $n$-dimensional left vector space over a division ring $R$. We write ${\mathcal G}_{k}(V)$ for the Grassmannian formed by $k$-dimensional subspaces of $V$ and denote by $\Gamma_{k}(V)$ the associated Grassmann graph. Let also…

This article is about a decoding algorithm for error-correcting subspace codes. A version of this algorithm was previously described by Rosenthal, Silberstein and Trautmann. The decoding algorithm requires the code to be defined as the…

The bisymplectic Grassmannian I$_2$Gr$(k, V)$ parametrizes k-dimensional subspaces of a vector space V which are isotropic with respect to two general skew-symmetric forms; it is a Fano variety which admits an action of a torus with a…

Let $\Pi$ be a polar space of rank $n\ge 3$. Denote by ${\mathcal G}_{k}(\Pi)$ the polar Grassmannian formed by singular subspaces of $\Pi$ whose projective dimension is equal to $k$. Suppose that $k$ is an integer not greater than $n-2$…

There are two widely used models for the Grassmannian $\operatorname{Gr}(k,n)$, as the set of equivalence classes of orthogonal matrices $\operatorname{O}(n)/(\operatorname{O}(k) \times \operatorname{O}(n-k))$, and as the set of trace-$k$…

The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems,…

We establish that every embedding of a Grassmann graph in a polar Grassmann graph can be reduced to an embedding in a Grassmann graph or to an embedding in the collinearity graph of a polar space. Also, we consider $3$-embeddings, i.e.…