We consider the category \(\mathcal {S}(n)\) of pairs \(X = (U,V)\) , where V is a finite-dimensional vector space with a nilpotent operator T with \(T^n = 0\) , and U is a subspace of V such that \(T(U) \subseteq U.\) For any vector space V, let |V| denote its dimension (or length). Note that \(\mathcal {S}(n)\) is just the category of Gorenstein-projective \(T_2(\Lambda )\) -modules, where \(\Lambda = k[T]/\langle T^n\rangle \) and \(T_2(\Lambda )\) is the ring of upper triangular \((2\times 2)\) -matrices with coefficients in \(\Lambda \) . We consider three related invariants for the objects X in \(\mathcal {S}(n)\) , the mean qX, the level pX and the colevel rX. 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 \(\mathcal {S}(n)\) , both numbers \(pX,\ rX\) are non-negative and \(pX + rX = qX \le n\) . It is the pr-triangle \(\mathbb {T}(n)\) of vectors (p, r) with \(p\ge 0,\ r\ge 0,\ p+r \le n,\) which we want to study in order to overview the category \(\mathcal {S}(n)\) . If X is an indecomposable object in \(\mathcal {S}(n)\) , we call \((pX,rX) \in \mathbb {T}(n)\) its support. We use \(\mathbb {T}(n)\) to visualize part of the categorical structure of \(\mathcal {S}(n)\) : The action of the duality \(\operatorname {D}\) and of the square \(\tau _n^2\) of the Auslander–Reiten translation are represented on \(\mathbb {T}(n)\) by a reflection and by a rotation by \(120^\circ \) , respectively. Moreover for \(n\ge 6\) , each component of the Auslander–Reiten quiver of \(\mathcal {S}(n)\) has support either contained in the center of \(\mathbb {T}(n)\) or with the center as its only accumulation point. We show that the only indecomposable objects X in \(\mathcal {S}(n)\) with support having boundary distance smaller than 1 are the pickets which lie on the boundary, whereas any rational vector in \(\mathbb {T}(n)\) 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 \(\mathbb {T}(n)\) provides even in the (quite well-understood) case \(n = 6\) some surprises: We will show that any indecomposable object in \(\mathcal {S}(6)\) lies on one of 12 central lines in \(\mathbb {T}(6)\) and that the center of \(\mathbb {T}(6)\) is the only vector which supports infinitely many indecomposables of \(\mathcal {S}(6)\) . 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 \(\mathbb {T}(n)\) , 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.
