PUB - Publikationen an der Universität Bielefeld

These notes are devoted to a single invariant, the Gabriel-Roiter measure of finite length modules: this invariant was introduced by Gabriel (under the name ‘Roiter measure’) in 1972 in order to give a combinatorial interpretation of the induction scheme used by Roiter in his 1968 proof of the first Brauer-Thrall conjecture. It is strange that this invariant (and Roiter’sproof itself) was forgotten in the meantime. One explanation may be that both Roiter and Gabriel pretend that their considerations are restricted to algebras of bounded representation type which are shown to be of finite representation type, thus restricted to algebras of finite representation type. But, as we are going to show, this invariant is of special interest when dealing with algebras of infinite representation type! And there may be a second explanation: in the early seventies, it was possible to calculate this invariant only for few examples, whereas nowadays there is a wealth of methods available. Looking at such examples, we are convinced that the Gabriel-Roiter measure has to be considered as a very important invariant and that it can be used to lay the foundation of the representation theory of artin algebras.