Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- Generalized hexagons and BLT-sets
- Orthogonally divergent spreads of Hermitian curves
- Lifts of nuclei in finite projective spaces
- Large minimal blocking sets, strong representative systems, and partial unitals
- The complement of a geometric hyperplane in a generalized polygon is usually connected
- Locally co-Heawood graphs
- A theorem of Parmentier characterizing projective spaces by polarities
- Geometries with diagram (diagram omitted)
- Remarks on finite generalized hexagons and octagons with a point-transitive automorphism group
- Block-transitive t-designs, II: large t
- Generalized Fischer spaces
- Ovoids and windows in finite generalized hexagons
- Flag transitive L.C2 geometries
- On nonics, ovals and codes in Desarguesian planes of even order
- Orbits of arcs in projective spaces
- There exists no (76,21,2,7) strongly regular graph
- Group-arcs of prime power order on cubic curves
- Planar Singer groups with even order multiplier groups
- On a footnote of Tits concerning Dn-geometries
- The structure of the central units of a commutative semifield plane
- Partially sharp subsets of PΓL(n, q)
- Partial ovoids and generalized hexagons
- A census of known flag-transitive extended grids
- Root lattice constructions of ovoids
- Coxeter groups in Coxeter groups
- A local characterization of the graphs of alternating forms
- A local characterization of the graphs of alternating forms and the graphs of quadratic forms over GF(2)
- On some locally 3-transposition graphs
- Coherent configurations derived from quasiregular points in generalized quadrangles
- Veldkamp planes
- The Lyons group has no distance-transitive representation
- Intersection of arcs and normal rational curves in spaces of odd characteristic
- Flocks and partial flocks of the quadratic cone in PG(3, q)
- Some extended generalized hexagons
- Nuclei in finite non-Desarguesian projective planes
Orbits of arcs in projective spaces
Published online by Cambridge University Press: 07 September 2010
- Frontmatter
- Contents
- Preface
- Introduction
- Generalized hexagons and BLT-sets
- Orthogonally divergent spreads of Hermitian curves
- Lifts of nuclei in finite projective spaces
- Large minimal blocking sets, strong representative systems, and partial unitals
- The complement of a geometric hyperplane in a generalized polygon is usually connected
- Locally co-Heawood graphs
- A theorem of Parmentier characterizing projective spaces by polarities
- Geometries with diagram (diagram omitted)
- Remarks on finite generalized hexagons and octagons with a point-transitive automorphism group
- Block-transitive t-designs, II: large t
- Generalized Fischer spaces
- Ovoids and windows in finite generalized hexagons
- Flag transitive L.C2 geometries
- On nonics, ovals and codes in Desarguesian planes of even order
- Orbits of arcs in projective spaces
- There exists no (76,21,2,7) strongly regular graph
- Group-arcs of prime power order on cubic curves
- Planar Singer groups with even order multiplier groups
- On a footnote of Tits concerning Dn-geometries
- The structure of the central units of a commutative semifield plane
- Partially sharp subsets of PΓL(n, q)
- Partial ovoids and generalized hexagons
- A census of known flag-transitive extended grids
- Root lattice constructions of ovoids
- Coxeter groups in Coxeter groups
- A local characterization of the graphs of alternating forms
- A local characterization of the graphs of alternating forms and the graphs of quadratic forms over GF(2)
- On some locally 3-transposition graphs
- Coherent configurations derived from quasiregular points in generalized quadrangles
- Veldkamp planes
- The Lyons group has no distance-transitive representation
- Intersection of arcs and normal rational curves in spaces of odd characteristic
- Flocks and partial flocks of the quadratic cone in PG(3, q)
- Some extended generalized hexagons
- Nuclei in finite non-Desarguesian projective planes
Summary
Abstract
An M × N matrix is associated with each ordered k-arc in a finite projective space of order N (k = M + N + 2.) The matrix is a projective invariant for ordered k-arcs in the space. The set of these matrices is denoted by Ω. Elements of the symmetric group Sk act on ordered arcs by permuting points. This induces a definition of Sk as a group of operators on Ω, whose orbits correspond to projectively distinct unordered k-arcs. Application of theorems of Burnside and Cauchy leads to results concerning the number of orbits of k-arcs in PG(N, q) under projectivity and under collineation. A subset of Ω is defined which contains representatives of each orbit under Sk. The reduced set of “normal” matrices is used by a counting algorithm. The results of this paper are applied to counting the projectively distinct unordered k-arcs (all k) in PG(2, 11) and PG(2, 13).
Introduction
This paper is concerned with the problem of counting equivalence classes of ordered k-arcs, unordered k-arcs, and k-gons with respect to either projectivity or collineation. Taking into account the three types of arc and two equivalence relations, there are six distinct but related problems to be solved. The six problems can each be stated in terms of the orbits of ordered arcs under some group of operators. The solutions, themselves, fall into two categories: formulas for the number of orbits and algorithms for counting the orbits.
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- Finite Geometries and Combinatorics , pp. 161 - 174Publisher: Cambridge University PressPrint publication year: 1993
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