The Deligne-Simpson problem for zero index of rigidity

Details The Deligne-Simpson problem for zero index of rigidity The Deligne-Simpson problem for zero index of rigidity

2000-11-16

arxiv.org/abs/math/0011107v1

Persp Cpx Anal, Proc. St Konstantin 2000 (2001) 1-35

Mathematics

Algebraic Geometry

To appear in the proceedings of the Fifth International Worksho p on Analysis, Differential geometry, Mathematical Physics and

Scientific paper

We consider the {\em Deligne-Simpson 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 matrices-residua 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 non-existence of such irreducible $(p+1)$-tuples.

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