Finite groups, smooth invariants, and isolated quotient singularities

2023/08/29 by Amiram Braun, Braun, Amiram
Mathematics · #Finite Group Theory Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.2308.15593

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.

Related

• Singularity categories and singular loci of certain quotient singularities
• p-Groups and the Polynomial Ring of Invariants Question
• Failure of F-purity and F-regularity in certain rings of invariants
• Three-dimensional isolated quotient singularities in odd characteristic
• Some invariant subalgebras are graded isolated singularities