The shift orbits of the graded Kronecker modules

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Abstract

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 \(n\ge 3\). In this paper, we usually will assume that \(n\ge 3\). 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 \({\text {mod}}(T(n),\Omega )\) 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 \(\sigma \mathpunct {:}{\text {mod}}(T(n),\Omega ) \rightarrow {\text {mod}}(T(n),\sigma \Omega )\) (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 \(\sigma ^tM \ne 0\) for all \(t\in \mathbb Z\). If pq are vertices of T(n), we denote by d(pq) 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 \(0 \le d(p(M),q(M)) \le r_0(M)\) 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 \(\sigma \)-orbit of M contains a unique sink module \(M_0\) with smallest possible radius, say with radius \(r_0 = r_0(M)\). For \(i\in \mathbb Z\), we write \(M_i = \sigma ^iM_0\) and call \(i = \iota (M_i)\) the index of \(M_i.\) By duality, the \(\sigma \)-orbit of M contains a unique source module with radius \(r_0\), say \(M_{b+1}\), and we have \(b\ge 0\). Let p(M) be the center of \(M_0\), let q(M) be the center of \(M_{b+1}\), and denote by \((p = a_0,a_1,\dots ,a_{b-1},a_b = q)\) the unique path from p to q. For \(i\ge 0\), the module \(M_{-i}\) is a sink module with center p(M) and radius \(r_0+i\), whereas the module \(M_{b+1+i}\) is a source module with center q(M) and radius \(r_0+i\). The remaining modules \(M_i\) (with \(1 \le i \le b\)) 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 pq of T(n) with \(0 \le d(p,q) \le r_0\) 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 \(M\rightarrow M'\), then we show that \(\iota (M') = \iota (M) -1\). In this way, we obtain a global way to index the regular indecomposable modules.

Keywords

Kronecker module Regular tree Matrix pencil 

Mathematics Subject Classification

Primary 16G20 16G70 15A22 Secondary 05C20 16G60 

Notes

Acknowledgements

The essential parts of the paper were presented in lectures at Shanghai and Beijing in 2015. The author is grateful to comments by mathematicians who attended these lectures. The author also wants to thank Daniel Bissinger, Philip Fahr, Rolf Farnsteiner and Otto Kerner for many fruitful discussions concerning the behavior of Kronecker modules, and the referee for a careful reading of the paper.

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Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany

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