Riemann-Roch定理是数学中的一个重要结论,并有了广泛的应用.在有限图和边加权有限图等图中也有对应的Riemann-Roch定理以及应用,但所有这些工作都有一个共同点,那就是它们都聚焦于在除子或和除子线性等价的线丛的情况下,也就是秩为1的情况.为 了得到高维秩的情形,可以借助多重除子的术语来描述.本文利用还原群GLn的root datum的概念给出了边加权有限图上主GLn-丛——向量丛的定义,并用多重除子的术语来描述向量丛,进而给出了边加权有限图的 Weil-Riemann-Roch定理以及证明,推广了 GROSS A.ULIRSCH M.和ZAKHAROV D 的结果.
Weil-Riemann-Roch theorem for edge-weighted finite graphs
The Riemann-Roch theorem is an important conclusion in mathematics and has been widely applied.There are al-so corresponding Riemann-Roch theorems and applications in finite graphs and edge-weighted finite graphs,but all of these works have a common point,which is that they focus on the case of a divisor or a line bundle equivalent to a divisor,that is,the case of rank 1.In the higher-rank situation,we have a similar description in terms of so-called multidivisors.In this paper,we mainly obtain the Weil-Riemann-Roch theorem for edge-weighted finite graphs and its proof.During this process,we define the principle GLn-bundles,which are vector bundles on edge-weighted finite graphs,by using the concept of the root datum of a re-ductive group GLn,and we use the term of multidivisors to describe vector bundles.This theorem generalizes the result of Gross A.Ulirsch M.and Zakharov D.
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