On the Deligne-Simpson problem

Details On the Deligne-Simpson problem On the Deligne-Simpson problem

2000-11-02

arxiv.org/abs/math/0011013v1

Proc. Steklov Inst. v. 238 (2002)

Mathematics

Algebraic Geometry

Scientific paper

The Deligne-Simpson problem is formulated like this: give necessary and sufficient conditions for the choice of the conjugacy classes $C_j\subset SL(n,{\bf C})$ or $c_j\subset sl(n,{\bf C})$ so that there exist irreducible $(p+1)$-tuples of matrices $M_j\in C_j$ or $A_j\in c_j$ satisfying the equality $M_1... M_{p+1}=I$ or $A_1+... +A_{p+1}=0$. We solve the problem for generic eigenvalues with the exception of the case of matrices $M_j$ when the greatest common divisor of the numbers $\Sigma_{j,l}(\sigma)$ of Jordan blocks of a given matrix $M_j$, with a given eigenvalue $\sigma$ and of a given size $l$ (taken over all $j$, $\sigma$, $l$) is $>1$. Generic eigenvalues are defined by explicit algebraic inequalities. For such eigenvalues there exist no reducible $(p+1)$-tuples. 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.

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