On the periodicity of Coxeter transformations and the non-negativity of their Euler forms

Abstract

We show that for piecewise hereditary algebras, the periodicity of the Coxeter transformation implies the non-negativity 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 give a simple, direct proof, that certain products of reflections, defined for any square matrix A with 2 on its main diagonal, and in particular the Coxeter transformation corresponding to a generalized Cartan matrix, can be expressed as $-A_{+}^{-1}{A}_{-}^t$, where A_{+}, A_{-} are closely associated with the upper and lower triangular parts of A.

arXiv comment12 pages, (v2) revision, to appear in Linear Algebra and its Applications

References in corpus (1)

• On Derived Equivalences of Categories of Sheaves Over Finite Posets

Cited by in corpus (5)

• Periodic trivial extension algebras and fractionally Calabi-Yau algebras
• The Coxeter transformation on Cominuscule Posets
• Categorification of a linear algebra identity and factorization of Serre functors
• Refined Coxeter polynomials
• Algebras whose Coxeter polynomials are products of cyclotomic polynomials