On some aspects of the Deligne-Simpson problem
The Deligne-Simpson problem is to find necessary and sufficient conditions for some conjugacy classes \(C_{j}\in GL_{n}({\mathbb C})\), so that there are irreducible matrices \(M_{j}\in C_{j}\) satisfying \(M_{1}\ldots M_{p+1}=I\). There is also an additive version of the Deligne-Simpson problem, studied by the author in [Proc. Steklov Inst. Math. 238, No. 3, 148-185 (2002) and Tr. Mat. Inst. Steklova 238, 158-195 (2002; Zbl 1036.34106)]. In the paper under review the author give a solution in the multiplicative case, when all the numbers of Jordan blocks of a matrix \(M_{j}\) which correspond to a same eigenvalue and a given size are divisible by a natural number \(d>1\). Generic eigenvalues, for which there exist no reducible solutions, are given by some inequalities. The matrices \(M_{j}\) which are solutions are interpreted as monodromy operators of regular linear systems on the Riemann sphere.
scientific article; zbMATH DE number 1971087
scientific article