图的局部因子化诱导的分支覆盖的谱与特征空间
Spectra and eigenspaces of branched covers of graphs induced by local factorisation
AI总结：
本文针对图的因子化提升（一类由局部因子化诱导的分支覆盖），利用群表示和复群环矩阵，提出了确定其完整谱的方法，并给出了特征向量提升的充分条件。
摘要：
我们考虑将图的特征值和特征向量提升到其{\em 因子化提升}，因子化提升是由非分支覆盖通过顶点稳定子子群诱导的所谓“局部因子化”形成的分支覆盖。这种描述适用于所有情形，其中图的分支覆盖的自同构群在覆盖的弧集上自由作用。具有此性质的分支覆盖可以通过群中的{\em 组合电压赋值}来描述，该赋值推广了正则覆盖理论中已知的（普通）电压赋值，并且是Potočnik和Toledo（2021）中图覆盖的代数描述的一个特例。借助群表示以及与待分支覆盖的图相关联的复群环上的某些矩阵，我们开发了一种确定因子化提升的完整谱的方法，并推导出提升特征向量的充分条件。
英文摘要：
We consider lifting eigenvalues and eigenvectors of graphs to their {\em factored lifts}, which are branched covers formed from unbranched ones by what one can describe as a `local factorisation' induced by subgroups of vertex stabilisers. Such a description applies to all situations in which a group of automorphisms of a branched cover of a graph acts freely on the arc set of the cover. Branched covers with this property can be described by means of {\em combined voltage assignments} in a group, which generalise the (ordinary) voltage assignments known from the theory of regular covering and are a particular case of an algebraic description of graph covers in Potočnik and Toledo (2021). With the help of group representations and certain matrices over complex group rings associated with the graphs to be branch-covered, we develop a method for determining the complete spectra of factored lifts, and derive a sufficient condition for lifting eigenvectors.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
