边加权有限图的Weil-Riemann-Roch定理  

Weil-Riemann-Roch theorem for edge-weighted finite graphs

作　　者：曹廷彬[1] 刘洁 CAO Tingbin;LIU Jie(Department of Mathematics of Nanchang University,Nanchang 330031,China)

机构地区：[1]南昌大学数学系,江西南昌330031

出　　处：《南昌大学学报（理科版）》2024年第2期103-107,共5页Journal of Nanchang University(Natural Science)

基　　金：国家自然科学基金资助项目(11871260);江西省自然科学基金资助项目(20232ACB201005)。

摘　　要：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。The Riemann-Roch theorem is an important conclusion in mathematics and has been widely applied.There are also 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 reductive 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.

关 键 词：边加权有限图 Riemann-Roch定理 向量丛 多重除子 

分 类 号：O157.5[理学—数学]