On the Harada and Sai bound

CHAIO CLAUDIA.

Goiania

Encuentro; Encuentro Sudamericano de Representaciones de Algebras; 2011

The concept of degree of an irreducible morphism was introduced by S. Liu [SL], in 1992. This notion has shown to be a very useful tool to solve many problems. In particular, by [CLMT] we are able to determine if a finite dimensional algebra over an algebraically closed field is of finite representation type computing the degree of a finite number of irreducible morphisms. The aim of this talk is to present some recent developments on degrees of irreducible morphisms. We are going to consider finite dimensional algebra over an algebraically closed field of finite representation type. It is well known that A is an artin algebra of finite representation type if and only if there exists a positive integer n such that Re^{n}(X,Y)=0 for all A-module X, Y. Moreover, by the Harada and Sai Lemma, we can consider n= 2^m-1 where m is the greater possible length of the indecomposable A-modules. In this talk, for a finite dimensional algebra over an algebraically closed field of finite representation type A, we are going to show a new bound n such that Re^n(X,Y)is not zero and Re^{n+1}(X,Y)=0 for all A-module X, Y. This bound is given in terms of degrees of irreducible morphisms. On the other hand, for such algebras we started to study how to read the degree of irreducible morphisms from its ordinary quiver. In particular, we will compute the degrees of irreducible morphisms in some string algebras of finite representation type. [CLMT] C. Chaio, P. Le Meur, S. Trepode. Degrees of the irreducible morphisms and finite-representation type. arXiv:0911.2296. To appear in J. London Math Soc. (2010) [SL] S. Liu. Degrees of irreducible maps and the shapes of Auslander-Reiten quivers. J. London Math. Soc (2) 45, (1992) 32-54.