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,
TrG: k[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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