PUB - Publikationen an der Universität Bielefeld

Let k be a field. The Kronecker modules (or matrix pencils) are the representations of the n-Kronecker quiver K(n); this is the quiver with two vertices, namely a sink and a source, and n arrows. The representations of K(n) play an important role in many parts of mathematics. For n = 2, the indecomposable representations have been classified by Kronecker, but not much is known in case n >= 3. In this paper, we usually will assume that n >= 3. The universal cover of K(n) is the n-regular tree with bipartite orientation. Let T (n) be the n-regular tree. We fix a bipartite orientation Omega of T (n); the opposite orientation will be denoted by sigma Omega (thus sigma(2)Omega = Omega). The k-representations of the quiver (T (n), Omega) can be considered as graded Kronecker modules and we denote by mod(T (n), Omega) the category of these graded Kronecker modules. Only few Kronecker modules can be graded, but the graded Kronecker modules provide hints about the behavior of general Kronecker modules. There is a reflection functor sigma: mod(T (n), Omega) -> mod(T (n), sigma Omega) (the simultaneous Bernstein-Gelfand-Ponomarev reflection at all sinks); it will be called the shift functor. An indecomposable graded Kronecker module M is said to be regular provided sigma M-t not equal 0 for all t is an element of Z. If p, q are vertices of T (n), we denote by d(p, q) their distance. Now, let M be an indecomposable regular representation of (T (n), Omega). We attach to M a positive integer r(0)(M) and a pair p(M), q(M) of vertices of T (n) with 0 <= d(p(M), q(M)) <= r(0)(M) and such that p(M) is a sink if and only if r(0)(M) is even. Here are the essential properties of the invariants r(0)(M), p(M), q(M). The s-orbit of M contains a unique sink module M-0 with smallest possible radius, say with radius r(0) = r(0)(M). For i is an element of Z, we write M-i = sigma(i) M-0 and call i = i(Mi) the index of Mi. By duality, the sigma-orbit of M contains a unique source module with radius r(0), say Mb+1, and we have b >= 0. Let p(M) be the center of M-0, let q(M) be the center of Mb+1, and denote by (p = a(0), a(1),..., a(b- 1), a(b) = q) the unique path from p to q. For i >= 0, the module M-i is a sink module with center p(M) and radius r(0) + i, whereas the module Mb+ 1+ i is a source module with center q(M) and radius r(0) + i. The remaining modules Mi (with 1 <= i <= b) are flow modules with radius r(0) - 1, and center {a(i-1), a(i)}. By construction, the triple r(0)(M), p(M), q(M) is invariant under the shift. We show that any triple r(0), p, q consisting of a positive integer r(0), and vertices p, q of T (n) with 0 <= d(p, q) <= r(0) and such that p a sink if and only if r(0) is even, arises in this way. If M, M' are regular indecomposable modules with an irreducible map M -> M', then we show that t(M') =t(M) - 1. In this way, we obtain a global way to index the regular indecomposable modules.