EMBEDDINGS OF COMPLEX LINE SYSTEMS AND FINITE REFLECTION GROUPS
Identifieur interne : 000783 ( Main/Exploration ); précédent : 000782; suivant : 000784EMBEDDINGS OF COMPLEX LINE SYSTEMS AND FINITE REFLECTION GROUPS
Auteurs : Muraleedaran Krishnasamy [Australie, Émirats arabes unis] ; D. E. Taylor [Australie]Source :
- Journal of the Australian Mathematical Society [ 1446-7887 ] ; 2008.
Abstract
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.
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DOI: 10.1017/S1446788708000955
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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.</div></front></TEI><affiliations><list><country><li>Australie</li><li>Émirats arabes unis</li></country><region><li>Nouvelle-Galles du Sud</li></region><settlement><li>Sydney</li></settlement><orgName><li>Université de Sydney</li></orgName></list><tree><country name="Australie"><region name="Nouvelle-Galles du Sud"><name sortKey="Krishnasamy, Muraleedaran" sort="Krishnasamy, Muraleedaran" uniqKey="Krishnasamy M" first="Muraleedaran" last="Krishnasamy">Muraleedaran Krishnasamy</name></region><name sortKey="Taylor, D E" sort="Taylor, D E" uniqKey="Taylor D" first="D. E." last="Taylor">D. E. Taylor</name><name sortKey="Taylor, D E" sort="Taylor, D E" uniqKey="Taylor D" first="D. E." last="Taylor">D. E. Taylor</name><name sortKey="Taylor, D E" sort="Taylor, D E" uniqKey="Taylor D" first="D. E." last="Taylor">D. E. 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