关于一个乘积等于单位元的代数方程的研究

A study on one algebraic equation of the product equaling unit

摘要:

本文考虑Deligne-Simpson问题和Hurwitz计数问题的关系. 首先观察到它们是不同群G上的相同代数方程:(A_1,B_1)…(A_g,B_g)X_1…X_k=I 的求解问题. 当G为复数域上的一般线性群时, 该方程等价于Deligne-Simpson问题; 当G为有限域上的一般线性群时, 该方程等价于Deligne-Simpson问题解空间欧拉数问题; 而当G为置换群时, 该方程等价于所谓的Hurwitz计数问题. 然后我们计算具有任意拆分的3 阶Deligne-Simpson 问题的欧拉特征类, 并将其中一些特征类的生成函数表示成了有理函数.

Abstract:

In this paper, we consider the relationship between the Deligne-Simpson problem and the Hurwitz enumeration problem. First, we observe that they are the solutions of the same algebraic equation on different groups, which the algebraic equation is (A_1,B_1)…(A_g,B_g)X_1…X_k=I. When G is the general linear group over the complex field, this equation is equivalent to Deligne-Simpson problem; When G is the general linear group over the finite field, this equation is equivalent to the Euler characteristic of solution space for Deligne-Simpson problem; When G is the permutation group, the equation is equivalent to the Hurwitz enumeration problem. Then we calculate the Euler characteristic of the 3th order Deligne-Simpson problem with any partition, and we express the generating function of some Euler characteristic as the rational functions.