Effective generation of rings of invariants of finite groups

Mufit Sezer, Purdue University

Abstract

We consider linear representations of a finite group G on a finite dimensional vector space over a field F. By a theorem due to E. Noether in char 0, and to Fleischmann and Fogarty in general, the ring of invariants is generated by homogeneous elements of degree at most :G: when :G: is invertible in F . Schmid, Domokos, and Hegedűs sharpened Noether's bound when G is not cyclic and char F = 0. In Chapter 1 we prove that the sharpened bound holds over general fields: If G is not cyclic and :G: is invertible in F, then the ring of invariants is generated by elements of degree at most ¾. :G: if :G: is even, and at most ⅝. :G: if :G: is odd. In Chapter 2 we consider the situation when G permutes a basis of V. Göbel proved that for n ≥ 3 the ring of invariants SG is generated by homogeneous elements of degree at most [special characters omitted]. For n ≥ 4 we sharpen this bound when further information on the action of G is available: If G is transitive but not 2-homogeneous, then SG is generated by elements of degree at most [special characters omitted] + 2. If G is j-homogeneous, but not (j + 1)-homogeneous, then SG is generated by elements of degree at most [special characters omitted]. We also prove that if G is cyclic of order n ≥ 4, then the invariants of the regular action are generated by elements of degree at most [special characters omitted] if n is even and [special characters omitted] if n is odd.

Degree

Ph.D.

Advisors

Avramov, Purdue University.

Subject Area

Mathematics

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