Some examples of rigid representations

Vladimir Kostov

paper

Some examples of rigid representations

arXiv:math/0006021

Abstract

Consider the Deligne-Simpson problem: {\em give necessary and sufficient conditions for the choice of the conjugacy classes $C_{j} \subset G L (n, C)$ (resp. $c_{j} \subset g l (n, C)$ ) so that there exist irreducible $(p + 1)$ -tuples of matrices $M_{j} \in C_{j}$ (resp. $A_{j} \in c_{j}$ ) satisfying the equality $M_{1} ... M_{p + 1} = I$ (resp. $A_{1} + ... + A_{p + 1} = 0$ )}. The matrices $M_{j}$ and $A_{j}$ are interpreted as monodromy operators and as matrices-residua of fuchsian systems on Riemann's sphere. We give new examples of existence of such $(p + 1)$ -tuples of matrices $M_{j}$ (resp. $A_{j}$ ) which are {\em rigid}, i.e. unique up to conjugacy once the classes $C_{j}$ (resp. $c_{j}$ ) are fixed. For rigid representations the sum of the dimensions of the classes $C_{j}$ (resp. $c_{j}$ ) equals $2 n^{2} - 2$ .