Examples illustrating some aspects of the weak Deligne-Simpson problem (Q2752989)

The weak Deligne-Simpson problem states: Give necessary and sufficient conditions for the conjugacy classes \(C_j\subset GL(n,{\mathbb C})\) such that there exist \((p+1)\)-tuples of matrices \(M_j\in C_j\) with trivial centralizers satisfying the equation \(M_1\ldots M_{p+1}=I\). The additive version of the problem considers the conjugacy classes \(c_j\subset gl(n,{\mathbb C})\) and matrices \(A_j\in c_j\) satisfying the condition \(A_1+\ldots+A_{p+1}=0\). The matrices \(M_j\) and \(A_j\) can be interpreted as monodromy operators and as matrices-residua of Fuchsian systems (i.e. with logarithmic poles) of \(n\) linear differential equations on the Riemann sphere. NEWLINENEWLINENEWLINEThe author considers the variety of \((p+1)\)-tuples of matrices belonging to fixed conjugacy classes and satisfying the weak Deligne-Simpson condition. He constructs new examples of such varieties and discusses their stratified structure and dimension of the strata. It turns out that the dimension of the varieties is higher than the expected, due to the presence of \((p+1)\)-tuples with non-trivial centralizers. In one of the examples the difference between the two dimensions is \({\mathcal O}(n)\).