Insights on Periodicity and Nonnegativity from Finite Dimensional Algebras

We show that for piecewise hereditary algebras, the periodicity of the Coxeter transformation implies the nonnegativity of the Euler form. Contrary to previous assumptions, the condition of piecewise heredity cannot be omitted, even for triangular algebras, as demonstrated by incidence algebras of posets. We also find that the only condition on a matrix A needed to express certain products of reflections defined by A as −A + A t − where A+, A− are closely associated with its upper and lower triangular parts, is that A should have 2 on its main diagonal.