Definitions and Notation

Noether Numbers:
Suppose that V is a representation of an algebraic group, G, over a field, k. Suppose that the ring of invariants k[V]G is finitely generated. This is guarenteed to happen if G is linearly reductive or if G is finite, for example. Let f1,f2,...,fr be a homogeneous minimal set of algebra generators for the algebra k[V]G. The number max{deg(fi) | i=1,2,...,r} is called the Noether Number of the representation and is denoted by b(k[V]G) and by b(V,G). More generally if R is any finitely generated graded algebra we denote by b(R) the largest degree of a generator in a homogeneous minimal generating set. More generally still, if I is a homogeneous ideal in a Noetherian algebra we denote by b(I) the largest degree of a generator in a homogeneous minimal generating set for I.
If G is a finite group then we have two possibilities: either the representation is modular or it is non-modular. If the characteristic of k is a positive prime, p, and if p divides the order of G then we say the representation is modular; otherwise it is non-modular. In 1916 Emmy Noether proved that if the characteristic of k is zero then b(V,G) <= |G|. Recently, P. Fleishmann and J. Forgarty independently proved that for any non-modular representation b(V,G) <= |G|. It is well known that b(V,G) may exceed |G| for modular representations.

If G is a finite group, we define the Transfer (also called Trace) homomorphism, TrGk[V] --> k[V]G by TrG(f):=S g e G (gf). Similarly we define the norm of a variable x as N(x):=P g e G (gx). For non-modular representations, the map TrG is surjective; for modular representations it is not onto and its image, Im TrG, is a non-zero proper ideal in k[V]G.
Related to the transfer is the notion of orbit sums. Given an element f of k[V] we consider its orbit under the action of G: {gf | g  e G} =: {f=f1, f2, ..., ft}. The orbit sum of f is the element O(f) := f1 + f2 + ... + ft. Clearly O(f) is an invariant. Moreover TrG(f) is an positive integer multiple of O(f).
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