Some examples of rigid representations (Q2731892)

The Deligne-Simpson problem states: Give necessary and sufficient conditions for the conjugacy classes \(C_j\subset GL(n,{\mathbb C})\) such that there exist irreducible \((p+1)\)-tuples of matrices \(M_j\in C_j\) 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. NEWLINENEWLINENEWLINEIn the paper under review the author gives new series of examples of \((p+1)\)-tuples of matrices satisfying the Deligne-Simpson condition. These examples are rigid, i.e. unique up to conjugacy once the classes \(C_j\) (respectively \(c_j\)) are fixed. In the rigid case the sum of the dimensions of the classes \(C_j\) (respectively \(c_j\)) is equal to \(2n^2-2\).