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

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We consider the category S(n) of 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. For any vector space V, let |V| denote its dimension (or length). Note that S(n) is just the category of Gorenstein-projective T2(Λ)-modules, where Λ=k[T]/⟨Tn⟩ and T2(Λ) is the ring of upper triangular (2×2)-matrices with coefficients in Λ. We consider three related invariants for the objects X in S(n), the meanqX, the levelpX and the colevelrX. By definition, qX=|V|/bV, pX=|U|/bV, and rX=|V/U|/bV. Here, bV denotes the dimension of the kernel of the operator T, thus the number of its Jordan blocks; we call bV the width of V. The objects X with bX=1 are called pickets. For any X in S(n), both numbers pX,rX are non-negative and pX+rX=qX≤n. It is the pr-triangle T(n) of vectors (p, r) with p≥0,r≥0,p+r≤n, which we want to study in order to overview the category S(n). If X is an indecomposable object in S(n), we call (pX,rX)∈T(n) its support. We use T(n) to visualize part of the categorical structure of S(n): The action of the duality D and of the square τn2 of the Auslander–Reiten translation are represented on T(n) by a reflection and by a rotation by 120∘, 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 the pickets 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; several partial results are included in the paper. The use of T(n) provides even in the (quite well-understood) case n=6 some surprises: We will show that any indecomposable object in S(6) lies on one of 12 central lines in T(6) and that the center of T(6) is the only vector which supports infinitely many indecomposables of S(6). A further target of our investigations is to single out settings which are purely combinatorial: this concerns not only the behaviour near the boundary of the triangle T(n), but also sets of indecomposable objects: for example, the pickets, the bipickets, as well as the objects X=(U,V) with U being cyclic. The paper is essentially self-contained, all prerequisites which are needed are outlined in detail.