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Geometric structures on finite- and infinite-dimensional Grassmannians

Andrea Blunck, Hans Havlicek

Abstract

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 (lines of the Grassmann space) and with so-called Z-reguli. We analyse the interdependencies among these different structures.

Geometric structures on finite- and infinite-dimensional Grassmannians

Abstract

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 (lines of the Grassmann space) and with so-called Z-reguli. We analyse the interdependencies among these different structures.

Paper Structure

This paper contains 5 sections, 17 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Grassmann graph and Grassmann space
  3. Distant graph and chain geometries
  4. $Z$-Reguli
  5. Consequences

Key Result

Lemma 2.1

Let ${\mathcal{G}}[M\rangle$ be a star. Then the following hold:

Theorems & Definitions (36)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • proof
  • Remark 2.6
  • ...and 26 more