Summary
Let G be a finite group acting linearly on the vector space V over a field of arbitrary characteristic. The action is called coregular if the invariant ring is generated by algebraically independent homogeneous invariants and the direct summand property holds if there is a surjective k[V]G-linear map π : k[V]→k[V]G. The following Chevalley–Shephard–Todd type theorem is proved. Suppose V is an irreducible kG-representation, then the action is coregular if and only if G is generated by pseudo-reflections and the direct summand property holds.
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Mathematics Subject Classification (2000): 13A50, 20H15
To Gerald Schwarz on the occasion of his 60th anniversary
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References
M. Aschbacher, Finite group theory, Second edition, Cambridge Studies in Advanced Mathematics, 10. Cambridge University Press, Cambridge, 2000.
D.J. Benson, Polynomial invariants of finite groups, London Mathematical Society Lecture Note Series, 190. Cambridge University Press, Cambridge, 1993.
A. Broer, The direct summand property in modular invariant theory, Transformation Groups 10 (2005) 5–27.
A. Broer, Invariant theory of abelian transvection groups, to appear in Canadian Mathematical Bulletin, preprint in arXiv:0709.0712 (math.AC).
H. Derksen and G. Kemper, Computational invariant theory, Invariant Theory and Algebraic Transformation Groups, I. Encyclopaedia of Mathematical Sciences, 130. Springer-Verlag, Berlin, 2002
G. Kemper and G. Malle, The finite irreducible linear groups with polynomial ring of invariants, Transformation Groups 2 (1997) 57–89.
H. Nakajima, Invariants of finite groups generated by pseudo-reflections in positive characteristic, Tsukuba J. Math. 3 (1979), 109–122.
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Broer, A. (2010). On Chevalley–Shephard–Todd’s Theorem in Positive Characteristic. In: Campbell, H., Helminck, A., Kraft, H., Wehlau, D. (eds) Symmetry and Spaces. Progress in Mathematics, vol 278. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4875-6_2
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DOI: https://doi.org/10.1007/978-0-8176-4875-6_2
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