Characterization of simple Coxeter-Dynkin diagrams (Q1916676)

There is a well-known system of relations that selects the simple Coxeter-Dynkin diagrams among the graphs with integral weights on the edges. Namely, the weights must be nonpositive and the corresponding (Killing) quadratic form must be positive definite. The latter condition can be written as a system of inequalities (the leading principal minors of the matrix of the quadratic form must be positive). In this note we give another system of equations and inequalities that also selects the diagrams \(A_n\), \(D_n\), \(E_6\), \(E_7\), and \(E_8\). The key point is the application of formulas for traces of the first and the second powers of the Coxeter operator. The Coxeter-Dynkin diagrams of singularities are graphs with integral weights on the edges, and we hope that the above-mentioned relationships can serve as a prototype for solving in the positive modality the following interesting problem: to determine the standard Coxeter-Dynkin diagram of a singularity (or a class of standard diagrams).