## Abstract

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 *T* : *V* → *V* a linear operator with *T ^{n}
* = 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 suffcient 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 indecomposable 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.

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