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4 - Combinatorial structures in finite classical polar spaces

Published online by Cambridge University Press:  21 July 2017

By
Antonio Cossidente
Edited by
Anders Claesson ,
Mark Dukes ,
Sergey Kitaev ,
David Manlove  and
Kitty Meeks
Antonio Cossidente
Affiliation:
Dipartimento di Matematica Informatica ed Economia
Anders Claesson
Affiliation:
University of Iceland, Reykjavik
Mark Dukes
Affiliation:
University College Dublin
Sergey Kitaev
Affiliation:
University of Strathclyde
David Manlove
Affiliation:
University of Glasgow
Kitty Meeks
Affiliation:
University of Glasgow
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Summary

Abstract

Sabsome recent results on regular systems and intriguing sets of finite classical polar spaces are surveyed.

Introduction

The finite classical polar spaces are the geometries associated with non–degenerate reflexive sesquilinear and non–singular quadratic forms on vector spaces of finite dimension over a finite field GF(q). Let PG(n, q) denote the n–dimensional projective space over GF(q). A polar space in PG(n, q) consists of its projective subspaces that are totally isotropic with respect to a given non–degenerate reflexive sesquilinear form or that are totally singular with respect to a given non–singular quadratic form. The rank of a polar space is the vector space dimension of a maximal totally isotropic or totally singular subspace, called here maximal. In this paper, the term polar space always refers to a finite classical polar space. A polar space of rank two is a generalised quadrangle. In the last decades, intensive investigations on combinatorial structures in finite polar spaces, such as spreads, ovoids, blocking sets, covers, have been carried out. More recently, other structures, such as m–systems, m–ovoids, i–tight sets (intriguing sets) have been studied. In this paper, some recent results on regular systems and intriguing sets of finite polar spaces are surveyed, with special emphasis on Hermitian polar spaces, that is, polar spaces arising from a non–degenerate unitary form.

Throughout the paper the following notation is adopted for finite polar spaces:

  1. (1)H(n, q2) is the space associated with a non–degenerate hermitian form on a vector space of dimension n + 1 over GF(q2);

  2. (2)Q-(n, q) is the space associated with a non –singular quadratic form of non–maximal Witt index on a vector space of dimension n+1 even over GF(q);

  3. (3)Q+(n, q) is the space associated with a non–singular quadratic form of maximal Witt index on a vector space of even dimension n+1 over GF(q);

  4. (4)Q(n, q) is the space associated with a non–singular quadratic form on a vector space of odd dimension n + 1 over GF(q);

  5. (5)W(n, q) is the space associated with a non–degenerate symplectic form on a vector space of even dimension n + 1 over GF(q).

Type
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Surveys in Combinatorics 2017 , pp. 204 - 237
Publisher: Cambridge University Press
Print publication year: 2017

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