Effective generation of rings of invariants of finite groups.

摘要

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 Hegedus 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. Gobel proved that for n ≥ 3 the ring of invariants SG is generated by homogeneous elements of degree at most n 2 . 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 n-1 2 + 2. If G is j-homogeneous, but not (j + 1)-homogeneous, then SG is generated by elements of degree at most n 2 - n-j-1 2 . 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 n2+ 2n-4 4 if n is even and n2+ 2n-3 4 if n is odd.