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

2002/07/31 by Friedrich Knop
Mathematics · #math.RT #math.AC #msc:13A50

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published as Invariant theory in all characteristics, 175-188, CRM Proc. Lecture Notes, 35, Amer. Math. Soc., Providence, RI, 2004 · 16 pages, v2: minor changes, final version

arxiv created 2003/10/07 · arxiv updated 2009/11/30

Abstract

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.

Related

• Weyl's polarization theorem in positive characteristic
• Vector invariants in arbitrary characteristic
• Vinberg's θ-groups in positive characteristic and Kostant-Weierstrass slices
• The finiteness theorem for invariants of a finite group (translation of Emmy Noether's "Der Endlichkeitsatz der Invarianten endlicher Gruppen")
• Noncommutative Noether's Problem vs Classical Noether's Problem