Methods to determine Coxeter polynomials
type: Item
instance of: scholarly article
title: Methods to determine Coxeter polynomials
zbMATH Open document ID: 0843.16011
DOI: 10.1016/0024-3795(94)00004-W
author: Axel Boldt
published in: Linear Algebra and its Applications
publication date: 1996-02-11T00:00:00Z
review text: Let \(\Lambda\) be a finite-dimensional algebra of finite global dimension. By a result of \textit{D. Happel} [Comment. Math. Helv. 62, 339-389 (1987; Zbl 0626.16008)] the derived category \(D^b(\Lambda)\) of bounded complexes over the category of finitely generated right \(\Lambda\)-modules has Auslander-Reiten triangles. The Auslander-Reiten translation yields on the level of the Grothendieck group \(K_0(\Lambda)\) an endomorphism \(\Phi_\Lambda\), called the Coxeter transformation. The author studies the Coxeter polynomial \(\text{det}(TE-\Phi_\Lambda)\). The first result is a reduction formula for the Coxeter polynomial \(\Phi_\Lambda\) if \(\Lambda\) is put together from subalgebras in a certain natural way. This yields in particular in the hereditary case explicit expressions for the Coxeter polynomial of large families of quivers. Furthermore, for some hereditary algebras which cannot be treated by this method the author gives explicit formulas for the Coxeter polynomial using a combinatorial interpretation of the entries of the Coxeter matrices.
reviewed by: Hagen Meltzer
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 16G30
zbMATH DE Number: 826713
zbMATH Keywords: finite-dimensional algebras of finite global dimension
zbMATH Keywords: categories of finitely generated right modules
zbMATH Keywords: Auslander-Reiten triangles
zbMATH Keywords: Auslander-Reiten translation
zbMATH Keywords: Grothendieck groups
zbMATH Keywords: Coxeter transformations
zbMATH Keywords: Coxeter polynomials
zbMATH Keywords: quivers
zbMATH Keywords: hereditary algebras
zbMATH Keywords: Coxeter matrices
zbMATH Keywords: derived categories
MaRDI profile type: Publication
cites work: Sur les valeurs propres de la transformation de Coxeter
cites work: On the derived category of a finite-dimensional algebra
cites work: Tame algebras and integral quadratic forms
Wikidata QID: Q126789304
description: scientific article; zbMATH DE number 826713