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April 10, 2015 By

The Noether Number in Invariant Theory

C. R. Math. Rep. Acad. Sci. Canada Vol. 28 (2) 2006, pp. 39–62

June 30, 2006

David L. Wehlau, Department of Mathematics & Computer Science, Royal Military College, Kingston, Ontario K7K 7B4; email: wehlau@rmc.ca

Abstract/Résumé:

Let \(\mathbb{F}\) be any field. Let \(G\) be any reductive linear algebraic group and consider a finite dimensional rational representation \(V\) of \(G\). Then the \(\mathbb{F}\)-algebra \(\mathbb{F}[V]^G\) of polynomial invariants for \(G\) acting on \(V\) is finitely generated. The Noether Number \(\beta(G,V)\) is the highest degree of an element of a minimal homogeneous generating set for \(\mathbb{F}[V]^G\). We survey what is known about Noether Numbers, in particular describing various upper and lower bounds for them. Both finite and infinite groups and both characteristic 0 and positive characteristic are considered.

Soit \(\mathbb{F}\) un corps commutatif. Soit \(G\) un groupe algébrique linéaire réductif, et \(V\) une représentation rationelle de dimension finie sur \(\mathbb{F}\). Alors \(\mathbb{F}[V]^G\), l’anneau des polynômes invariants pour l’action de \(G\) sur \(V\), admet un nombre fini de générateurs. Le nombre de Noether \(\beta(G,V)\) est le degré maximal d’un membre d’un ensemble minimal de générateurs homogènes de \(\mathbb{F}[V]^G\). Nous faisons une revue des résultats connus sur les nombres de Noether. En particulier, nous décrivons certaines bornes supérieures et inférieures pour les nombres de Noether. Nous considérons à la fois les groupes finis et infinis, sur des corps de charactéristique \(0\) ou \(p>0\).

Keywords: Noether number, degree bounds, invariant rings
AMS Subject Classification: Actions of groups on commutative rings; invariant theory 13A50

[This journal is open access except for the current year and the preceding 5 years]

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