Examples illustrating some aspects of the weak DeligneSimpson 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 DeligneSimpson 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 matricesresidua 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 nontrivial centralizers; in one of the examples the difference between the two dimensions is O(n).
 Publication:

arXiv Mathematics eprints
 Pub Date:
 January 2001
 arXiv:
 arXiv:math/0101141
 Bibcode:
 2001math......1141P
 Keywords:

 Algebraic Geometry;
 Rings and Algebras;
 Representation Theory
 EPrint:
 Research partially supported by INTAS grant 971644