Abstract
We prove the following result: LetG be a finite irreducible linear group. Then the ring of invariants ofG is a polynomial ring if and only ifG is generated by pseudoreflections and the pointwise stabilizer inG of any nontrivial subspace has a polynomial ring of invariants. This is well-known in characteristic zero. For the case of positive characteristic we use the classification of finite irreducible groups generated by pseudoreflections due to Kantor, Wagner, Zalesskiî and Serežkin. This allows us to obtain a complete list of all irreducible linear groups with a polynomial ring of invariants.
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References
D. J. Benson,Polynomial Invariants of Finite Groups, LMS Lecture Notes Series, vol. 190, Cambridge University Press, Cambridge,1993.
N. Bourbaki,Groupes et Algèbres de Lie, IV, V, VI, Hermann, Paris, 1968.
D. Carlisle and P. H. Kropholler,Rational invariants of certain orthogonal and unitary groups, Bull. London Math. Soc.24 (1992), 57–60.
D. Carlisle and P.H. Kropholler,Modular invariants of finite symplectic groups, preprint (1992).
C. Chevalley,Invariants of finite groups generated by reflections, Amer. J. Math77 (1955), 778–782.
A. M. Cohen,Finite complex reflection groups, Ann. Scient. Éc. Norm. Sup.9 (1976), 379–436.
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson,Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
M. Demazure,Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math.21 (1973), 287–301.
L. E. Dickson,A fundamental system of invariants of the general modular linear group with a solution of the form problem, Trans. Amer. Math. Soc.12 (1911), 75–98.
G.-M. Greuel, G. Pfister, H. Schönemann, SINGULAR,a Computer Algebra System for Singularity Theory, Algebraic Geometry and Commutative A Algebra, Universität Kaiserslautern, Germany.
V. Kac, K. Watanabe,Finite linear groups whose ring of invariants is a complete intersection, Bull. Amer. Math. Soc.6 (1982), 221–223.
W. Kantor,Subgroups of classical groups generated by long root elements, Trans. Amer. Math. Soc.248 (1979), 347–379.
G. Kemper,A constructive approach to Noether's problem, Manuscripta Math.90 (1996), 343–363.
G. Kemper,Calulating invariant rings of finite groups over arbitrary fields, J. Symbolic Computation21 (1996), 351–366.
H. Nakajima,Invariants of finite groups generated by pseudoreflections in positive characteristic, Tsukuba J. Math.3 (1979), 109–122.
M. Schönert (editor), GAP—Groups, Algorithms, and Programming, fourth ed., Lehrstuhl D für Mathematik, RWTH Aachen, Germany, 1994.
J.-P. Serre,Groupes finis d'automorphismes d'anneaux locaux réguliers, Colloque d'Algèbre ENSJF, Paris, 1967, pp. 8-01–8-11.
G. C. Shephard and J. A. Todd,Finite unitary reflection groups, Canad. J. Math.6 (1954), 274–304.
L. Smith,Polynomial Invariants of Finite Groups, AK Peters, Wellesley, 1995.
R. Steinberg,Differential equations invariant under finite reflection groups, Trans. Math. Soc.112 (1964), 392–400.
A. Wagner,Groups generated by elations, Abh. Math. Seminar Univ. Hamburg41 (1974), 190–205.
A. Wagner,Collineation groups generated by homologies of order greater than 2, Geom. Dedicata7 (1978), 387–398.
A. Wagner,Determination of the finite primitive reflection groups over an arbitrary field of characteristic not 2, I, Geom. Dedicata9 (1980), 239–253; II, ibid. Geom. Dedicata10, 183–189; III, Geom. Dedicata ibid.10, 475–523.
C. Xu,Computing invariant polynomials of p-adic reflection groups, Proc. Symp. Appl. Math., vol. 48, Amer. Math. Soc., Providence, 1994.
A. E. Zalesskiî and V. N. Serežkin,Linear groups generated by transvections, Izv. Akad. Nauk SSSR, Ser. Mat.40 (1976), no. 1, 26–49 (in Russian). English translation: A. E. Zalesskiî and V. N. Serežkin,Linear groups generated by transvections, Math. USSR Izv.10 (1976), 25–46.
A. E. Zalesskiî and V. N. Serežkin,Finite linear groups generated by reflections, Izv. Akad. Nauk SSSR, Ser. Mat.44 (1980), no. 6, 1279–1307 (in Russian). English translation: A. E. Zalesskiî and V. N. Serežkin,Finite linear groups generated by reflections, Math. USSR Izv.17 (1981), 477–503.
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The second author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft
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Kemper, G., Malle, G. The finite irreducible linear groups with polynomial ring of invariants. Transformation Groups 2, 57–89 (1997). https://doi.org/10.1007/BF01234631
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DOI: https://doi.org/10.1007/BF01234631