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EMBEDDINGS OF COMPLEX LINE SYSTEMS AND FINITE REFLECTION GROUPS

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

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

References

[1]Bourbaki, N., Groupes et algèbres de Lie, Chapitres 4, 5 et 6 (Hermann, Paris, 1968).Google Scholar
[2]Cameron, P. J., Goethals, J.-M., Seidel, J. J. and Shult, E. E., ‘Line graphs, root systems, and elliptic geometry’, J. Algebra 43(1) (1976), 305–327.CrossRefGoogle Scholar
[3]Cohen, Arjeh M., ‘Finite complex reflection groups’, Ann. Sci. Ecole Norm. Sup. (4) 9(3) (1976), 379–436.CrossRefGoogle Scholar
[4]Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of Finite Groups (Oxford University Press, Eynsham, 1985).Google Scholar
[5]Cvetković, Dragoš, Rowlinson, Peter and Simić, Slobodan, Spectral Generalizations of Line Graphs, London Mathematical Society Lecture Note Series, 314 (Cambridge University Press, Cambridge, 2004).CrossRefGoogle Scholar
[6]Delsarte, P., Goethals, J. M. and Seidel, J. J., ‘Bounds for systems of lines, and Jacobi polynomials’, Philips Res. Rep. 30 (1975), 91–105.Google Scholar
[7]Kantor, William M., ‘Generation of linear groups’, in: The Geometric Vein: The Coxeter Festschrift (eds. Chandler Davis, Branko Grünbaum and F. A. Sherk) (Springer, New York, 1981),pp. 497–509.CrossRefGoogle Scholar
[8]Koornwinder, Tom H., ‘A note on the absolute bound for systems of lines’, Indag. Math (N.S.) 38 (1976), 152–153.CrossRefGoogle Scholar
[9]Mitchell, H. H., ‘Determination of all primitive collineation groups in more than four variables which contain homologies’, Amer. J, Math. 36 (1914), 1–12.CrossRefGoogle Scholar
[10]Shephard, G. C. and Todd, J. A., ‘Finite unitary reflection groups’, Canad. J. Math. 6 (1954), 274–304.CrossRefGoogle Scholar