Periodic Coxeter matrices
Cartan \((C_A)\) and Coxeter \((\varphi_A)\) matrices are considered. Conditions are given on the eigenvalues of \(\varphi_A\), which imply the periodicity of \(\varphi_A\). If \(\varphi_A\) is periodic then the Euler form \(q_A(x)= xC_A^{-t}x^t\) is nonnegative. Examples are given. The relation of the Coxeter matrices \(\varphi_A\) and \(\varphi_B\) for a one-point extension \(A=B[M]\) of a \(k\)-algebra \(B\) by a \(B\)-module \(M\) is studied.
scientific article; zbMATH DE number 1915484