Related papers: On geometry of linear involutions

This note proves that, for $F = \Bbb{R,C}$ or $\Bbb{H}$, the bordism classes of all non-bounding Grassmannian manifolds $G_k(F^{n+k})$, with $k < n$ and having real dimension $d$, constitute a linearly independent set in the unoriented…

Let $k,l,m,n$ be positive integers such that $m-l\ge l>k, m-l>n-k\ge k$ and $m-l>2k^2-k-1$. Let $G_{k}(\mathbb{C}^n)$ denote the Grassmann manifold of $k$-dimensional vector subspaces of $\bc^n$. We show that any continuous map…

We show that (equivariant) K-theoretic 3-point Gromov-Witten invariants of genus zero on a Grassmann variety are equal to triple intersections computed in the ordinary (equivariant) K-theory of a two-step flag manifold, thus generalizing an…

We study the Grassmann manifold $G_k$ of all $k$-dimensional subspaces of ${\mathbb R}^n$. The Cartan embedding $G_k\subset O(n)$ realizes $G_k$ as a subspace of $Sl_n({\mathbb R})$ and we study the decomposition $G_k=\coprod_w (BwB\cap…

We pose and solve the equivalence problem for subspaces of ${\mathcal P}_n$, the $(n+1)$ dimensional vector space of univariate polynomials of degree $\leq n$. The group of interest is ${\rm SL}_2$ acting by projective transformations on…

In math.RT/0302174 we developed a framework to study representations of groups of the form $G((t))$, where $G$ is an algebraic group over a local field $K$. The main feature of this theory is that natural representations of groups of this…

The integral singular cohomology ring of the Grassmann variety parametrizing $r$-dimensional subspaces in the $n$-dimensional complex vector space is naturally an irreducible representation of the Lie algebra of all the $n\times n$ matrices…

Let $V$ be a linear representation of a connected complex reductive group $G$. Given a choice of character $\theta$ of $G$, Geometric Invariant Theory defines a locus $V^{ss}_\theta(G) \subseteq V$ of semistable points. We give necessary,…

Let $\sigma_1$ and $\sigma_2$ be commuting involutions of a semisimple algebraic group $G$. This yields a $Z_2\times Z_2$-grading of $\g=\Lie(G)$, $\g=\bigoplus_{i,j=0,1}\g_{ij}$, and we study invariant-theoretic aspects of this…

Let $\mathbb{Z}_{p^s}$ be the residue class ring of integers modulo $p^s$, where $p$ is a prime number and $s$ is a positive integer. Using matrix representation and the inner rank of a matrix, we study the intersection, join, dimension…

Let F be a non-Archimedean locally compact field with residual characteristic p, let G be an inner form of GL(n,F) for a positive integer n and let R be an algebraically closed field of characteristic different from p. When R has…

Let $G$ be a reductive group over a field $k$ of characteristic $\neq 2$, let ${\mathfrak g}=\Lie(G)$, let $\theta$ be an involutive automorphism of $G$ and let ${\mathfrak g}={\mathfrak k}\oplus{\mathfrak p}$ be the associated symmetric…

A simple Almost-Riemannian Structure on a Lie group G is defined by a linear vector field (that is an infinitesimal automorphism) and dim(G) -- 1 left-invariant ones. It is first proven that two different ARSs are isometric if and only if…

Let $H$ be an infinite-dimensional complex Hilbert space and let ${\mathcal G}_{\infty}(H)$ be the set of all closed subspaces of $H$ whose dimension and codimension both are infinite. We investigate (not necessarily surjective)…

Let $G_{n,k}$ denote the real Grassmann manifold of $k$-dimensional vector subspaces of $\mathbb R^n$. Using the Hodgkin spectral sequence, we compute the complex $K$-ring of $G_{n,k}$, up to a small indeterminacy, for all values of $n,k$…

In this paper first we describe all (not necessarily linear or bijective) transformations on $\mathbb{R}^d$ with $2\leq d<\infty$ which preserve the area of parallelograms spanned by any two vectors. We also characterize those (not…

A dual formulation of the S Matrix for N=4 SYM has recently been presented, where all leading singularities of n-particle N^{k-2}MHV amplitudes are given as an integral over the Grassmannian G(k,n), with cyclic symmetry, parity and…

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…

In this article we study the involutions of $\mathrm{O}(V,\mathrm{q})$, an orthogonal group for a vector space $V$ with quadratic form $\mathrm{q}$ over a field of characteristic 2. The classification proceeds by discussing conjugacy…

When $G$ is a complex reductive algebraic group and $G/K$ is a reductive symmetric space, the decomposition of $\C[G/K]$ as a $K$-module was obtained (in a non-constructive way) by Richardson, generalizing the celebrated result of…