Related papers: Morphisms between Grassmannians, II

Denote by $\mathbb G(k,n)$ the Grassmannian of linear subspaces of dimension $k$ in $\mathbb P^n$. We show that if $n>m$ then every morphism $\varphi: \mathbb G(k,n) \to \mathbb G(l,m)$ is constant.

We consider the Grassmann graph of $k$-dimensional subspaces of an $n$-dimensional vector space over the $q$-element field and its subgraph $\Gamma(n,k)_q$ formed by non-degenerate linear $[n,k]_q$ codes. We assume that $1<k<n-1$. It is…

Let $V$ be an $n$-dimensional left vector space over a division ring $R$ and $n\ge 3$. Denote by ${\mathcal G}_{k}$ the Grassmann space of $k$-dimensional subspaces of $V$ and put ${\mathfrak G}_{k}$ for the set of all pairs $(S,U)\in…

Let $\mathcal{F}_h^i(k,n)$ be the $i$th ordered configuration space of all distinct points $H_1,\ldots,H_h$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of $\mc^n$, whose sum is a subspace of dimension $i$. We prove that…

Given an $n$-dimensional vector space $V$ over a field $\mathbb K$, let $2\leq k < n$. There is a natural correspondence between the alternating $k$-linear forms $\varphi$ of $V$ and the linear functionals $f$ of $\bigwedge^kV$. Let…

The Grassmannians of lines in projective N-space, G(1,N), are embedded by way of the Pl"ucker embedding in the projective space P(\bigwedge^2 C^{N+1}). Let H^l be a general l-codimensional linear subspace in this projective space. We…

Consider the Grassmann graph of $k$-dimensional subspaces of an $n$-dimensional vector space over the $q$-element field, $1<k<n-1$. Every automorphism of this graph is induced by a semilinear automorphism of the corresponding vector space…

Let $\psi : G\to GL(V)$ and $\varphi :G \to GL (W)$ be representations of finite group $G$. A linear map $T: V\to W$ is called a morphism from $\psi$ to $\varphi$ if it satisfys $T\psi_g= \varphi_g T$ for each $g\in G$ and let…

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 $n-2$…

Let (M,g) be a compact Riemannian manifold of dimension n. For k \in {0,...,n}, we denote Gr_{k}(M) the set of compact, connected and oriented submanifolds of M of dimension k. This set is called the non-linear Grassmannian. In this…

We characterize all $(n-2)$-dimensional linear subspaces of $\mathbb{P}^{n}$ such that the induced linear projection, when restricted to the rational normal curve, gives a Galois morphism. We give an explicit description of these spaces as…

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…

Let $V$ be an $n$-dimensional vector space ($4\le n <\infty$) and let ${\mathcal G}_{k}(V)$ be the Grassmannian formed by all $k$-dimensional subspaces of $V$. The corresponding Grassmann graph will be denoted by $\Gamma_{k}(V)$. We…

Let $G_{n,k}$ denote the complex Grassmann manifold of $k$-dimensional vector subspaces of $\mathbb{C}^n$. Assume $l,k\le \lfloor n/2\rfloor$. We show that, for sufficiently large $n$, any continuous map $h:G_{n,l}\to G_{n,k}$ is rationally…

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 $\Gamma(n,k)$…

Integrable flows on the Grassmannians Gr(N-1,N+1) are defined by the requirement of closedness of the differential N-1 forms $\Omega_{N-1}$ of rank N-1 naturally associated with Gr(N-1,N+1). Gauge-invariant parts of these flows, given by…

There is a hierarchy of commuting soliton equations associated to each symmetric space U/K. When U/K has rank n, the first n flows in the hierarchy give rise to a natural first order non-linear system of partial diffferential equations in n…

Let $k \subset K$ be an extension of fields, and let $A \subset M_{n}(K)$ be a $k$-algebra. We study parameter spaces of $m$-dimensional subspaces of $K^{n}$ which are invariant under $A$. The space $\mathbb{F}_{A}(m,n)$, whose $R$-rational…

We construct a non-commutative version of the Grassmann variety $G(2,4)$ as a non-commutative moduli space of linear subspaces in a projective space.

In this paper, we study the Grassmannian of n-dimensional subspaces of a 2n-dimensional vector space and its infinite-dimensional analogues. Such a Grassmannian can be endowed with two binary relations (adjacent and distant), with pencils…