Periodic Coxeter matrices Original Research Article

Let A=kQ/I be a finite dimensional triangular k-algebra. Consider the Cartan matrix CA and the Coxeter matrix phiA=−CA−tCA. Let χphi(T)=det(Tid−phiA) be the Coxeter polynomial of A. We study conditions on Spec phiA in order that phiA is a periodic matrix. We show that in case phiA is periodic then the Euler quadratic form qA(x)=xCA−txt is non-negative and qA>0 if and only if 1negated set membership Spec phiA.