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Canonical Systems of Basic Invariants for Unitary Reflection Groups

Published online by Cambridge University Press:  20 November 2018

Norihiro Nakashima ,
Hiroaki Terao  and
Shuhei Tsujie
Norihiro Nakashima
Affiliation:
School of Information Environment, Tokyo Denki University, Inzai, 270-1382, Japan e-mail: nakashima@mail.dendai.ac.jp
Hiroaki Terao
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan e-mail: hterao00@za3.so-net.ne.jp e-mail: tsujie@math.sci.hokudai.ac.jp
Shuhei Tsujie
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan e-mail: hterao00@za3.so-net.ne.jp e-mail: tsujie@math.sci.hokudai.ac.jp
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Abstract

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It is known that there exists a canonical system for every finite real reflection group. In a previous paper, the first and the third authors obtained an explicit formula for a canonical system. In this article, we first define canonical systems for the finite unitary reflection groups, and then prove their existence. Our proof does not depend on the classification of unitary reflection groups. Furthermore, we give an explicit formula for a canonical system for every unitary reflection group.

Keywords

13A5020F55basic invariantinvariant theoryfinite unitary reflection group

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 3 , 01 September 2016 , pp. 617 - 623
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Bourbaki, N., Groupes et Algèbres de Lie. Chapitres 4, 5, et 6, Hermann, Paris, 1968.Google Scholar
[2] Chevalley, C., Invariants of finite groups generated by reflections. Amer. J. Math, 77(1955), no. 4, 778–782. http://dx.doi.org/10.2307/2372 597 Google Scholar
[3] Flatto, L., Basic sets of invariants for finite reflection groups. Bull. Amer. Math. Soc. 74(1968), no. 4, 730–734. http://dx.doi.org/10.1090/S0002-9904-1968-12017-8 Google Scholar
[4] Flatto, L., Invariants of finite reflection groups and mean value problems. II. Amer. J. Math. 92(1970), 552–561. http://dx.doi.org/10.2307/2373360 Google Scholar
[5] Flatto, L. and Wiener, M. M., Invariants of finite reflection groups and mean value problems. Amer. J. Math. 91(1969), no. 3, 591–598. http://dx.doi.org/10.2307/2373340 Google Scholar
[6] Humphreys, J. E., Reflection groups and Coxeter groups. Cambridge Studies in Mathematics, 29, Cambridge University Press, Cambridge, 1990.Google Scholar
[7] Iwasaki, K., Basic invariants of finite reflection groups. J. Algebra. 195(1997), no. 2, 538–547. http://dx.doi.org/10.1006/jabr.1 997.7066 Google Scholar
[8] Kane, R., Reflection groups and invariant theory. CMS Books in Mathematics/Ouvragesde Mathématiques de la SMC, 5, Springer-Verlag, New York, 2001. http://dx.doi.org/=10.1007/978-l-4757-3542-0 Google Scholar
[9] Nakashima, N. and Tsujie, S., A canonical system of basic invariants of a finite reflection group. J. Algebra 406(2014), 143–153. http://dx.doi.org/10.101 6/j.jalgebra.2O14.02.012 Google Scholar
[10] Orlik, P. and Solomon, L., Unitary reflection groups and cohomology. Invent. Math. 59(1980), no. 1, 77–94. http://dx.doi.org/10.1007/BF01390316 Google Scholar
[11] Orlik, P. and Terao, H., Arrangements of hyperplanes. Grundlehrendermatematischen Wissenschaften, 300, Springer-Verlag, Berlin, 1992.Google Scholar
[12] Steinberg, R., Differential equations invariant under finite reflection groups. Trans. Amer. Math. Soc. 112(1964), no. 3, 392–400. http://dx.doi.org/10.1090/S0002-9947-1964-0167535-3 Google Scholar