The Noether Bound in Invariant Theory of Finite Groups

Fleischmann, Peter

doi:10.1006/aima.2000.1952

Cited by 67 publications

(48 citation statements)

References 14 publications

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“…In Noether (1916) she showed that if the characteristic of k is zero then S G is always generated by the invariant polynomials in S of degree less than or equal to |G|. Recently this result has been extended independently by Fleischmann (2000) and Fogarty (2001) to the general case of a non-modular representation. The result does not hold for modular representations.…”

Section: Introductionmentioning

confidence: 87%

Computing Modular Invariants of p-groups

Shank

Wehlau

2002

Journal of Symbolic Computation

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Let V be a finite dimensional representation of a p-group, G, over a field, k, of characteristic p. We show that there exists a choice of basis and monomial order for which the ring of invariants, k[V ] G , has a finite SAGBI basis. We describe two algorithms for constructing a generating set for k[V ] G . We use these methods to analyse k[2V 3 ] U 3 where U 3 is the p-Sylow subgroup of GL 3 (Fp) and 2V 3 is the sum of two copies of the canonical representation. We give a generating set for k[2V 3 ] U 3 for p = 3 and prove that the invariants fail to be Cohen-Macaulay for p > 2. We also give a minimal generating set for k[mV 2 ] Z/p were V 2 is the two-dimensional indecomposable representation of the cyclic group Z/p.

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Section: Introductionmentioning

confidence: 87%

Computing Modular Invariants of p-groups

Shank

Wehlau

2002

Journal of Symbolic Computation

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“…After a first draft of this note has been completed it came to my attention that this has also been done independently by Fleischmann,[Fl1], and Sezer, [Se]. Nevertheless, we find it still worthwhile to include a proof.…”

Section: Introductionmentioning

confidence: 87%

On Noether’s and Weyl’s bound in positive characteristic

Knop¹

2004

CRM Proceedings and Lecture Notes

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In this note we generalize several well known results concerning invariants of finite groups from characteristic zero to positive characteristic not dividing the group order. The first is Schmid's relative version of Noether's theorem. That theorem compares the degrees of generators of a group with those of a subgroup. Then we prove a suitable positive characteristic version of Weyl's theorem on vector invariants: polarization works in small degrees. Using that we show that the regular representation has the "most general" ring of invariants, thereby generalizing theorems of Schmid and Smith.

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“…Recently a very short proof for the nonmodular case was found by Fleischmann [5] and Fogarty [6]. Their proof can be slightly modified so that it holds in the generality stated above.…”

Section: The Noether Boundmentioning

confidence: 99%

The Direct Summand Property in Modular Invariant Theory

Broer

2005

Transformation Groups

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In nonmodular invariant theory of finite groups, the invariant ring is always a direct summand of the full polynomial ring. This is no longer generally true in modular invariant theory, but nevertheless interesting examples are known where this happens. We give useful characterisations of the direct summand property in terms of the image of a twisted transfer map. For example, for p-groups acting in characteristic p the direct summand property holds if and only if the image of the ordinary transfer is a principal ideal; and in that case the group is generated by transvections. We also extend some known results in the nonmodular case to where only the direct summand property is assumed, e.g., the invariant ring is always generated by its elements of degree at most the order of the group.

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The Noether Bound in Invariant Theory of Finite Groups

Cited by 67 publications

References 14 publications

Computing Modular Invariants of p-groups

Computing Modular Invariants of p-groups

On Noether’s and Weyl’s bound in positive characteristic

The Direct Summand Property in Modular Invariant Theory

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