Some bounds for the spectral radius of a coxeter transformation
M. Takane, la Pena J.A. de · Tsukuba Journal of Mathematics · 1993
Let $\Delta$ be a finite quiver ( $=oriented$ , connected graph) without oriented cycles.Let $k$ be any field.The path algebra $k[\Delta]$ is a hereditary algebra, see [7].The study of this kind of algebras had played a central role in the development of the Representation Theory of Algebras, see [6,4,13,11].For a representation $X$ of $k[\Delta]$ , we denote by $\underline{\dim}X=(\dim_{k}X(i))_{t\in\Delta_{0}}$ the dimension vector of $X$ , where $\Delta_{0}$ is the set of vertices of $\Delta$ .The Coxeterwhere $\tau X$ denotes the Auslander-Reiten translate of the non-projective indecom- posable representation $X$ .The spectral radius $\rho(\phi_{\Delta})$ of the Coxeter matrix $\phi_{\Delta}$ , contains relevant information about the behaviour of the translation $\tau$ , see [5,11].In this work, we consider some elementary relations between the spectral radii $\rho(\phi_{E})$ and $\rho(\phi_{\Delta})$ for a Galois covering $\pi:\overline{\Delta}\rightarrow\Delta$ .In particular, we show that for any covering $\pi:\overline{\Delta}\rightarrow\Delta$ defined by the action of a residually finite group and any finite subgraph $F$ of $\overline{\Delta}$ , we have $\rho(\phi_{F})\leqq\rho(\phi_{\Delta})$ .In [12], we have explored the relations between the spectral radii $r(\Delta)$ and $r(\overline{\Delta})$ of the adjacency matrices $A_{\overline{\Delta}}$ and $A_{\Delta}$ , for a Galois covering $\pi;\overline{\Delta}\rightarrow\Delta$ .In section 2, we show how to use these results to get some interesting bounds for $\rho(\phi_{\Delta})$ .Finally, we get some applications.In relation with a problem posed by Kerner, we show that $\frac{g(\Delta)}{\rho(\phi_{\Delta})}\leqq\frac{|\Delta_{0}|}{2}$ , where $g(\Delta)=|\Delta_{1}|-|\Delta_{0}|+1$ denotes the genus of the underlying graph of $\Delta$ .1. Galois covering and Coxeter matrices.1.1.Let $n$ be the number of vertices of the quiver $\Delta$ .