Related papers: On geometry of linear involutions

We determine the group of linear transformations on a vector space $V$ that preserve a polynomial function $f$ on $V$ for several interesting pairs $(V,f)$, using the theory of semisimple algebraic groups.

Let $\Delta$ be a thick building of type $\textsf{X}_{n}=\textsf{C}_{n},\textsf{D}_{n}$. Let also ${\mathcal G}_k$ be the Grassmannian of $k$-dimensional singular subspaces of the associated polar space $\Pi$ (of rank $n$). We write…

We introduce a method to determine if n-dimensional smooth subvarieties of an ambient space of dimension at most 2n − 2 inherit the Picard group from the ambient space (as it happens when the ambient space is a projective space,…

Let $V$ and $W$ be finite dimensional real vector spaces and let $G\subset\GL(V)$ and $H\subset\GL(W)$ be finite subgroups. Assume for simplicity that the actions contain no reflections. Let $Y$ and $Z$ denote the real algebraic varieties…

Let $n,k$ denote integers with $n>2k\geq 6$. Let $\mathbb{F}_q$ denote a finite field with $q$ elements, and let $V$ denote a vector space over $\mathbb{F}_q$ that has dimension $n$. The projective geometry $P_q(n)$ is the partially ordered…

We study linear projections on Pluecker space whose restriction to the Grassmannian is a non-trivial branched cover. When an automorphism of the Grassmannian preserves the fibers, we show that the Grassmannian is necessarily of…

Let U be the tautological subbundle on the Grassmannian $\mathrm{Gr}(k, n)$. There is a natural morphism $\mathrm{Tot}(U) \to \mathbb{A}^n$. Using it, we give a semiorthogonal decomposition for the bounded derived category…

Our goal is to find classes of convolution semigroups on Lie groups $G$ that give rise to interesting processes in symmetric spaces $G/K$. The $K$-bi-invariant convolution semigroups are a well-studied example. An appealing direction for…

Let K be a field of characteristic 2 and G a nonabelian locally finite 2-group. Let V(KG) be the group of units with augmentation 1 in the group algebra KG. An explicit list of groups is given, and it is proved that all involutions in V(KG)…

Let $k$ be a field with characteristic different from $2$. In this paper, we describe the $k$-rational orbit spaces in some irreducible prehomogeneous vector spaces $(G,V)$ over $k$, where $G$ is a connected reductive algebraic group…

We consider the action of the one-parameter subgroup of the special linear group corresponding to a simple root on Grassmannians and describe the structure of the associated Geometric Invariant Theory (GIT) quotients with respect to…

Let $G$ be a connected, simply connected three-dimensional Lie group (unimodular or non-unimodular) equipped with a left-invariant (Riemannian or Lorentzian) metric $g$. By definition, the isometry group $\mathrm{Isom}(G, g)$ contains $G$…

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 $\Gamma(n,k)$…

Given a complex vector space $V$ of finite dimension, its Grassmannian variety parametrizes all subspaces of $V$ of a given dimension. Similarly, if a finite group $G$ acts on $V$, its invariant Grassmannian parametrizes all the…

Given a Schubert class on $Gr(k,V)$ where $V$ is a symplectic vector space of dimension $2n$, we consider its restriction to the symplectic Grassmannian $SpGr(k,V)$ of isotropic subspaces. Pragacz gave tableau formulae for positively…

The Johnson graph J(v,k) has, as vertices, the k-subsets of a v-set V, and as edges the pairs of k-subsets with intersection of size k-1. We introduce the notion of a neighbour-transitive code in J(v,k). This is a vertex subset \Gamma such…

We study the geometry of an important class of generic curves in the Grassmannian manifolds of $n$-dimensional subspaces and Lagrangian subspaces of $R^{2n}$ under the action of the linear and linear symplectic group.

Given an $N$-dimensional vector space $V$ over a field $\mathbb{F}$ and a trace-valued $(\sigma,\varepsilon)$-sesquilinear form $f:V\times V\rightarrow \mathbb{F}$, with $\varepsilon = \pm 1$ and $\sigma^2 = \mathrm{id}_{\mathbb{F}}$, let…

An automorphism of a graph $G$ with $n$ vertices is a bijective map $\phi$ from $V(G)$ to itself such that $\phi(v_i)\phi(v_j)\in E(G)$ $\Leftrightarrow$ $v_i v_j\in E(G)$ for any two vertices $v_i$ and $v_j$ of $G$. Denote by…

Let G be the group preserving a nondegenerate sesquilinear form on a vector space V, and H a symmetric subgroup of G of the type G1 x G2. We explicitly parameterize the H-orbits in the Grassmannian of r-dimensional isotropic subspaces of V…