A Database of Invariant Rings

Kemper, Gregor; Körding, Elmar; Malle, Gunter; Matzat, B. Heinrich; Vogel, Denis; Wiese, Gabor

doi:10.1080/10586458.2001.10504673

Cited by 11 publications

(5 citation statements)

References 13 publications

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“…Theorem 4 in [64] shows that A G is Gorenstein if the image of the associated map λ : G → GL(m A /m 2 A ) is in SL(m A /m 2 A ). See also Conjecture 5 in [29] when |G| is not invertible in A = k[V ], where V is a k-vector space with an action of G.…”

Section: Class Group and Canonical Classmentioning

confidence: 99%

Moderately ramified actions in positive characteristic

Lorenzini

Schröer

2019

Math. Z.

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In characteristic 2 and dimension 2, wild Z/2Z-actions on k [[u, v]] ramified precisely at the origin were classified by Artin, who showed in particular that they induce hypersurface singularities. We introduce in this article a new class of wild quotient singularities in any characteristic p > 0 and dimension n ≥ 2 arising from certain non-linear actions of Z/pZ on the formal power series ring k[[u 1 , . . . , u n ]]. These actions are ramified precisely at the origin, and their rings of invariants in dimension 2 are hypersurface singularities, with an equation of a form similar to the form found by Artin when p = 2. In higher dimension, the rings of invariants are not local complete intersection in general, but remain quasi-Gorenstein. We establish several structure results for such actions and their corresponding rings of invariants.

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Section: Class Group and Canonical Classmentioning

confidence: 99%

Moderately ramified actions in positive characteristic

Lorenzini

Schröer

2019

Math. Z.

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“…The following example illustrates this point as it results in various Cohen-Macaulay defects. It is taken from the database of invariant rings of Kemper et al [22]. Example 2.5.…”

Section: Polynomial Separating Algebrasmentioning

confidence: 99%

Separating invariants of finite groups

Reimers

2018

Journal of Algebra

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This paper studies separating invariants of finite groups acting on affine varieties through automorphisms. Several results, proved by Serre, Dufresne, Kac-Watanabe and Gordeev, and Jeffries and Dufresne exist that relate properties of the invariant ring or a separating subalgebra to properties of the group action. All these results are limited to the case of linear actions on vector spaces. The goal of this paper is to lift this restriction by extending these results to the case of (possibly) non-linear actions on affine varieties.Under mild assumptions on the variety and the group action, we prove that polynomial separating algebras can exist only for reflection groups. The benefit of this gain in generality is demonstrated by an application to the semigroup problem in multiplicative invariant theory.Then we show that separating algebras which are complete intersections in a certain codimension can exist only for 2-reflection groups. Finally we prove that a separating set of size n + k − 1 (where n is the dimension of X) can exist only for k-reflection groups.Several examples show that most of the assumptions on the group action and the variety that we make cannot be dropped.

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“…The following example is constructed of the same type and results in various Cohen-Macaulay defects. It is taken from Kemper's et al [9] database of invariant rings.…”

Section: Generated By K-reflectionsmentioning

confidence: 99%

Polynomial Separating Algebras and Reflection Groups

Reimers¹

2013

Preprint

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This note considers a finite algebraic group G acting on an affine variety X by automorphisms. Results of Dufresne on polynomial separating algebras for linear representations of G are extended to this situation. For that purpose, we show that the Cohen-Macaulay defect of a certain ring is greater than or equal to the minimal number k such that the group is generated by (k + 1)-reflections. Under certain rather mild assumptions on X and G we deduce that a separating set of invariants of the smallest possible size n = dim(X) can exist only for reflection groups.

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A Database of Invariant Rings

Cited by 11 publications

References 13 publications

Moderately ramified actions in positive characteristic

Moderately ramified actions in positive characteristic

Separating invariants of finite groups

Polynomial Separating Algebras and Reflection Groups

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