Examples illustrating some aspects of the weak Deligne-Simpson pro blem

Abstract

We consider the variety of $(p+1)$-tuples of matrices $A_j$ (resp. $M_j$) from given conjugacy classes $c_j\subset gl(n,{\bf C})$ (resp. $C_j\subset GL(n,{\bf C})$) such that $A_1+... +A_{p+1}=0$ (resp. $M_1... M_{p+1}=I$). This variety is connected with the weak {\em Deligne-Simpson problem: give necessary and sufficient conditions on the choice of the conjugacy classes $c_j\subset gl(n,{\bf C})$ (resp. $C_j\subset GL(n,{\bf C})$) so that there exist $(p+1)$-tuples with trivial centralizers of matrices $A_j\in c_j$ (resp. $M_j\in C_j$) whose sum equals 0 (resp. whose product equals $I$).} The matrices $A_j$ (resp. $M_j$) are interpreted as matrices-residua of Fuchsian linear systems (resp. as monodromy operators of regular linear systems) on Riemann's sphere. We consider examples of such varieties of dimension higher than the expected one due to the presence of $(p+1)$-tuples with non-trivial centralizers; in one of the examples the difference between the two dimensions is O(n).