Let k be a field. We consider triples (V,U,T), where V is a finite
dimensional k-space, U a subspace of V and TV→V a linear
operator with Tn=0 for some n, and such that T(U)⊆U. Thus,
T is a nilpotent operator on V, and U is an invariant subspace with
respect to T.
We will discuss the question whether it is possible to classify these
triples. These triples (V,U,T) are the objects of a category with the
Krull-Remak-Schmidt property, thus it will be sufficient to deal with
indecomposable triples. Obviously, the classification problem depends on n,
and it will turn out that the decisive case is n=6. For n<6, there are
only finitely many isomorphism classes of indecomposables triples, whereas for
n>6 we deal with what is called ``wild'' representation type, so no
complete classification can be expected.
For n=6, we will exhibit a complete description of all the indecomposable
triples.Comment: 55 pages, minor modification in (0.1.3), to appear in: Journal fuer
die reine und angewandte Mathemati