The DeligneSimpson problem for zero index of rigidity
Abstract
We consider the {\em DeligneSimpson problem}: {\em Give necessary and sufficient conditions for the choice of the conjugacy classes $c_j\subset gl(n,{\bf C})$ or $C_j\subset GL(n,{\bf C})$, $j=1,..., p+1$, so that there exist irreducible $(p+1)$tuples of matrices $A_j\in c_j$ whose sum is 0 or of matrices $M_j\in C_j$ whose product is $I$.} The matrices $A_j$ (resp. $M_j$) are interepreted as matricesresidua of Fuchsian linear systems (resp. as monodromy operators of regular systems) on Riemann's sphere. We consider the case when the sum of the dimensions of the conjugacy classes $c_j$ or $C_j$ is $2n^2$ and we prove a theorem of nonexistence of such irreducible $(p+1)$tuples.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 November 2000
 arXiv:
 arXiv:math/0011107
 Bibcode:
 2000math.....11107P
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Rings and Algebras;
 Mathematics  Representation Theory
 EPrint:
 To appear in the proceedings of the Fifth International Worksho p on Analysis, Differential geometry, Mathematical Physics and Applications (Complex Structures and Vector Fields), St. Constantine resort (near Varna, Bulgaria), September 4  12, 2000