2006/11/30 by Sefi Ladkani
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #math.CO #math.RT #msc:06A11 #msc:15A63 #msc:16G20
paper · pdf · doi:10.1016/j.laa.2007.08.002
published as Linear Algebra and its Applications 428 (2008), 742-753 · 12 pages, (v2) revision, to appear in Linear Algebra and its Applications
arxiv created 2007/08/07 · openalex publication_date 2007/09/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02
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.