A new relation among Cartan matrix and Coxeter matrix (Q1088736)

The object of the paper is to show a relation between invariants of a root system R of type \(A_{\ell}\) (\(\ell odd)\), \(D_{\ell}, E_ 6, E_ 7\), or \(E_ 8\). On the one hand there are invariants p, q, r, and d associated to a Cartan matrix of R as follows. Let d be the determinant of the Cartan matrix, and let p, q, r be the lengths of the branches of the corresponding Dynkin diagram (which is interpreted as branching at the middle vertex in the case \(A_{\ell}\), \(\ell odd)\). On the other hand a Coxeter transformation of R determines invariants a, b, c, and h as follows. Let h be the order of the Coxeter transformation C, and let \(m_ 1,...,m_{\ell}\) be the exponents of C, i.e. exp(2\(\pi\sqrt{- 1}m_ i/h)\) \((i=1,...,\ell)\) are the eigenvalues of C. Then there are positive integers a, b, and c such that \(T^{m_ 1}+T^{m_ 2}+...+T^{m_{\ell}}=T^{-h}(T^ h-T^ a)(T^ h-T^ b)(T^ h- T^ c)/(T^ a-1)(T^ b-1)(T^ c-1)\). The author shows the numerical identity: \(pqr/d=abc/h.\) The proof uses algebraic geometric facts concerning the rational double point corresponding to the root system R. After multiplication by four, both sides of the equation can be identified with the order of the binary polyhedral group associated to the rational singularity.