Methods to determine Coxeter polynomials
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.
finite-dimensional algebras of finite global dimension
categories of finitely generated right modules
scientific article; zbMATH DE number 826713