, Volume 2, Issue 1, pp 57-89

The finite irreducible linear groups with polynomial ring of invariants

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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.

The second author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft