A study on one algebraic equation of the product equaling unit

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.