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EMBEDDINGS OF COMPLEX LINE SYSTEMS AND FINITE REFLECTION GROUPS

Part of: Special aspects of infinite or finite groups Metric geometry Real and complex geometry

Published online by Cambridge University Press:  01 October 2008

MURALEEDARAN KRISHNASAMY  and
D. E. TAYLOR
MURALEEDARAN KRISHNASAMY
Affiliation:
School of Mathematics and Statistics, University of Sydney, Australia
D. E. TAYLOR*
Affiliation:
School of Mathematics and Statistics, University of Sydney, Australia (email: D.Taylor@maths.usyd.edu.au)
*
✉For correspondence; e-mail: D.Taylor@maths.usyd.edu.au
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Abstract

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A star is a planar set of three lines through a common point in which the angle between each pair is 60∘. A set of lines through a point in which the angle between each pair of lines is 60 or 90∘ is star-closed if for every pair of its lines at 60∘ the set contains the third line of the star. In 1976 Cameron, Goethals, Seidel and Shult showed that the indecomposable star-closed sets in Euclidean space are the root systems of types An, Dn, E6, E7 and E8. This result was a key part of their determination of all graphs with least eigenvalue −2. Subsequently, Cvetković, Rowlinson and Simić determined all star-closed extensions of these line systems. We generalize this result on extensions of line systems to complex n-space equipped with a hermitian inner product. There is one further infinite family, and two exceptional types arising from Burkhardt and Mitchell’s complex reflection groups in dimensions five and six. The proof is a geometric version of Mitchell’s classification of complex reflection groups in dimensions greater than four.

Keywords

systems of linesunitary reflection groupcomplex reflection grouppseudo-reflection

MSC classification

Secondary: 51F15: Reflection groups, reflection geometries 20F55: Reflection and Coxeter groups 51M20: Polyhedra and polytopes; regular figures, division of spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 85 , Issue 2 , October 2008 , pp. 211 - 228
Copyright
Copyright © 2008 Australian Mathematical Society

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