The Deligne-Simpson problem -- a survey

Vladimir Petrov Kostov

paper

The Deligne-Simpson problem -- a survey

arXiv:math/0206298

Abstract

The Deligne-Simpson problem (DSP) (resp. the weak DSP) is formulated like this: {\em give necessary and sufficient conditions for the choice of the conjugacy classes $C_{j} \subset G L (n, C)$ or $c_{j} \subset g l (n, C)$ so that there exist irreducible (resp. with trivial centralizer) $(p + 1)$ -tuples of matrices $M_{j} \in C_{j}$ or $A_{j} \in c_{j}$ satisfying the equality $M_{1} ... M_{p + 1} = I$ or $A_{1} + ... + A_{p + 1} = 0$ }. The matrices $M_{j}$ and $A_{j}$ are interpreted as monodromy operators of regular linear systems and as matrices-residua of Fuchsian ones on Riemann's sphere. The present paper offers a survey of the results known up to now concerning the DSP.