On the Auslander-Reiten Quiver with Oriented Cycles of Representation- nite Algebras

Let A be a basic, connected nite-dimensional algebra over an algebraically closed eld with nite representation type, (A) be the maximum of the set consisting of numbers of indecomposable summands in the middle term of Auslander-Reiten sequences in modA. In this paper I show that if (A)=1, then ? A contains oriented cycles if and only if ? A contains DTr-periodic modules. When (A) 2 I give counterexamples to the assertion. Let E be a skeletally small category with nite hom-sets. Then the generating function of E is a formal summation E(t) := X X2E= = 1 jAut(X)j t X : A (strict) KS-category E is a category with nite coproducts in which each object X has a strictly unique decomposition X = ` I into a nite coprod-uct of connected objects of E. We denote by Con(E) the full subcategory of connected objects. Then the exponential formula holds: Theorem. E(t) = exp(Con(E)(t)) for a KS-category E. There are many applications of this exponential formula, e.g., an enu-meration of group homomorphisms from a nite group to a symmetirc group S n , an enumeration of rooted trees. The our theory is closely related to Joyal's species: a species is bijectively correspondent with a faithful functor with nite bers from a groupoids to Set f. We can generalize operations on species to categories and faithful