Invariant Subspaces of Nilpotent Operators. Level, Mean, and Colevel: The Triangle T(n)

We consider the category S(n) of all pairs $X = (U,V)$, where V is a finite-dimensional vector space with a nilpotent operator T with Tⁿ = 0, and U is a subspace of V such that T(U) ⊆ U. Our main interest in an object $X=(U,V)$ are the three numbers uX= U (for the subspace), wX= V/U (for the factor) and bX= Ker T (for the operator). Actually, instead of looking at the reference space R³ with the triples $(uX,wX,bX)$, we will focus the attention to the corresponding projective space T(n) which contains for a non-zero object X the level-colevel pair { pr}$X = (uX/bX,wX/bX)$ supporting the object X. We use T(n) to visualize part of the categorical structure of S(n): The action of the duality D and the square τₙ² of the Auslander-Reiten translation are represented on T(n) by a reflection and a rotation by 120^∘ degrees, respectively. Moreover for n≥ 6, each component of the Auslander-Reiten quiver of S(n) has support either contained in the center of T(n) or with the center as its only accumulation point. We show that the only indecomposable objects X in S(n) with support having boundary distance smaller than 1 are objects with $bX=1$ which lie on the boundary, whereas any rational vector in T(n) with boundary distance at least 2 supports infinitely many indecomposable objects. At present, it is not clear at all what happens for vectors with boundary distance between 1 and 2. The use of T(n) provides even in the (quite well-understood) case $n = 6$ some surprises: In particular, we will show that any indecomposable object in S(6) lies on one of 12 central lines in T(6). The paper is essentially self-contained, all prerequisites which are needed are outlined in detail.