Examples Illustrating some Aspects of the Weak Deligne-Simpson Problem

Abstract

We consider the variety of (p + 1)-tuples of matrices Aj (resp. Mj ) from given conjugacy classes cj ⊂ gl(n, C) (resp. Cj ⊂ GL(n, C)) such that A1 + . . . + A[p+1] = 0 (resp. M1 . . . M[p+1] = I). This variety is connected with the weak Deligne-Simpson problem: give necessary and sufficient conditions on the choice of the conjugacy classes cj ⊂ gl(n, C) (resp. Cj ⊂ GL(n, C)) so that there exist (p + 1)-tuples with trivial centralizers of matrices Aj ∈ cj (resp. Mj ∈ Cj ) whose sum equals 0 (resp. whose product equals I). The matrices Aj (resp. Mj ) 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).