Some Examples Related to the Deligne–Simpson Problem

Abstract

We consider the variety of $(p+1)$-tuples of matrices $M_j$ from given conjugacy classes $C_j \subset \text{GL}(n,\mathbb{C})$ such that $M_1 \cdot M_{p+1}=I$. This variety is connected with the Deligne-Simpson problem: Give necessary and sufficient conditions on the choice of the conjugacy classes $C_j \subset \text{GL}(n,\mathbb{C})$ such that there exist irreducible $(p+1)$-tuples of matrices $M_j \in C_j$ whose product equals $I$. The matrices $M_j$ are interpreted as monodromy operators of regular linear systems on Riemann's sphere. We consider among others cases when the dimension of the variety is higher than the expected one due to the presence of $(p + 1)$-tuples with non-trivial centralizers.