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An Aspect of Icosahedral Symmetry

Published online by Cambridge University Press:  20 November 2018

Jan Rauschning  and
Peter Slodowy
Jan Rauschning
Affiliation:
Fachbereich Mathematik der Universität Hamburg Universität Hamburg D-20146 Hamburg Germany, e-mail: jan [email protected]
Peter Slodowy
Affiliation:
Fachbereich Mathematik der Universität Hamburg Universität Hamburg D-20146 Hamburg Germany, e-mail: [email protected]
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Abstract

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We embed the moduli space $Q$ of 5 points on the projective line ${{S}_{5}}$-equivariantly into $\mathbb{P}\left( V \right)$, where $V$ is the 6-dimensional irreducible module of the symmetric group ${{S}_{5}}$. This module splits with respect to the icosahedral group ${{A}_{5}}$ into the two standard 3-dimensional representations. The resulting linear projections of $Q$ relate the action of ${{A}_{5}}$ on $Q$ to those on the regular icosahedron.

Keywords

14L2420B25
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 45 , Issue 4 , 01 December 2002 , pp. 686 - 696
Copyright
Copyright © Canadian Mathematical Society 2002

References

[1] Barthel, G., Hirzebruch, F. and Höfer, Th., Geradenkonfigurationen und algebraische Flächen. Aspekte der Math. D4, Vieweg Verlag, 1987.Google Scholar
[2] Cohen, P. B. and Hirzebruch, F., Lecture notes of a graduate course at the ETH Zürich. 1996.Google Scholar
[3] Demazure, M., Surfaces de del Pezzo. In: Séminaire sur les Singularit és des surfaces, (eds., M. Demazure, H. Pinkham, B. Teissier), Lecture Notes in Math. 777, Berlin, Heidelberg, New York, 1980, 23–69.Google Scholar
[4] Hermite, Ch., Sur l'équation du cinquifieme degré. C.R.A.S. 61/62 (1865/66), Oeuvres t. II, 347–424, Gauthiers-Villars, Paris, 1905–1917.Google Scholar
[5] Hilbert, D., Theory of Algebraic Invariants. Lecture Notes of a course in Göttingen, 1897, Cambridge University Press, 1998.Google Scholar
[6] Holzapfel, G., Geometry and Arithmetic; Around Euler Partial Differential Equations. VEB Deutscher Verlag der Wissenschaften, Berlin, 1986.Google Scholar
[7] Kantor, S., Theorie der endlichen Gruppen von eindeutigen Transformationen in der Ebene. Mayer und Müller, Berlin, 1895.Google Scholar
[8] Klein, F., Vorlesungen über das Ikosaeder und die Gleichungen vom fünften Grade. Teubner, Leipzig, 1884, new edition, with commentaries by P. Slodowy, Birkhäuser, Basel, and Teubner, Stuttgart, 1993.Google Scholar
[9] Manin, Y., Cubic Forms: Algebra, Geometry, Arithmetic. 2nd ed., North Holland, Amsterdam, New York, 1986.Google Scholar
[10] Moore, E. H., The cross-ratio group of n! Cremona transformations of order n − 3 in flat spaces of n − 3 dimensions. Amer. J. Math. 22 (1900), 279–291.Google Scholar
[11] Mumford, D., Fogarty, J. and Kirwan, F., Geometric Invariant Theory. Ergebnisse derMathematik und ihrer Grenzgebiete 34, 3rd enlarged edition, Springer, Berlin, Heidelberg, New York, 1994.Google Scholar
[12] Mumford, D. and Suominen, K., Introduction to the theory of moduli. In: Proceedings of the 5th Nordic Summer School in Mathematics Algebraic Geometry, Oslo, 1970, (ed., F. Oort), Amer. J. Math. 77 (1955), 171–222.Google Scholar
[13] Rauschning, J., Eine birationale Aktion der Ikosaedergruppe. Diplomarbeit, Fachbereich Mathematik, Universität Hamburg, 2002.Google Scholar
[14] Renner, L., Binary quintics. Proccedings of the 1984 Vancouver conference in Algebraic Geometry. CanadianMath. Soc Conf. Proc. 6, 369–374, Amer. Math. Soc., Providence, R.I., 1986.Google Scholar
[15] Slaught, H. E., The cross-ratio group of 120 quadratic Cremona transformations of the plane I: Geometric Representation. Amer. J. Math. 22 (1900), 343–388.Google Scholar
[16] Szurek, M., Binary quintics and the icosahedron. Proccedings of the 1984 Vancouver conference in Algebraic Geometry. Canadian Math. Soc Conf. Proc. 6, Amer. Math. Soc., Providence, R.I., 1986, 473–475.Google Scholar
[17] Slodowy, P., Über das Ikosaeder und die Gleichungen fu¨nften Grades. In: Mathematische Miniaturen Band 3, Arithmetik und Geometrie, (eds., H. Knörrer, C.-G. Schmidt, J. Schwermer, P. Slodowy), Birkhäuser, Basel, 1986, 71–113.Google Scholar
[18] Swinnerton-Dyer, H. P. F., Rational points on del Pezzo surfaces of degree 5. In: Proceedings of the 5th Nordic Summer School in Mathematics Algebraic Geometry, Oslo, 1970, (ed., F. Oort), Amer. J. Math. 77 (1955), 287–290.Google Scholar
[19] Weyl, H., The Classical Groups. Princeton University Press, 1946.Google Scholar