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

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

2006-11-07

arxiv.org/abs/math/0611201v2

Linear Algebra and its Applications 428 (2008), 742-753

Mathematics

Representation Theory

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

Scientific paper

10.1016/j.laa.2007.08.002

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.

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