Finite groups, smooth invariants, and isolated quotient singularities
Abstract
Let G < SL(V) be a finite group, V is finite dimensional over a field F, p=char F and S(V) is the symmetric algebra of V. We determine when the subring of G-invariants S(V)^G is a polynomial ring. As a consequence, we classify, if F is algebraically closed, all S(V)^G which are isolated singularities. We show that the completion of S(V)^G, at its unique graded maximal ideal, is isomorphic to the completion of S(W)^H, where (H,W) is a reduction mod p of a member of the Zassenhaus-Vincent-Wolf list of complex isolated quotient singularities.
Keywords
Cite
@article{arxiv.2308.15593,
title = {Finite groups, smooth invariants, and isolated quotient singularities},
author = {Amiram Braun},
journal= {arXiv preprint arXiv:2308.15593},
year = {2024}
}
Comments
-"W is void of fixed points with respect to G/T(G)" is inserted in the introduction and proved in Thm. B. -Prop. 2.14 items (4),(5) added. -Thm. 2 case (i) is explained by adding decomposition series of H-modules. -The proof of Cor. 3.9 was wrong in cases SO(4,q)^(-), SO(6,q)^(-,+), q even. Kleidman-Liebeck tables were incorrectly used. This is corrected in Note 3.12 - Prop. 3.16