Helene Tyler, Manhattan College

Classification of Module Categories via the Gabriel-Roiter
Measure

This talk reflects joint work with Markus Schmidmeier

The Gabriel-Roiter measure of a module was introduced in 2005 by
C.M. Ringel. It is based on P. Gabriel's refinement of the induction
scheme that A. Roiter used in his proof of the first Brauer-Thrall
conjecture. The Gabriel-Roiter measure provides a tripartite
classification mechanism for the category of modules over a
finite-dimensional algebra, alternative to the traditional
classification offered by the Auslander-Reiten theory. In earlier
work, we undertook a detailed study of the Gabriel-Roiter measures
for tame hereditary quivers of type $\widetilde{\mathbb A}_n$, and
their interplay with the Auslander-Reiten theory. In our current
work, we expand our purview to a larger class of algebras for which
the Auslander-Reiten theory is well understood, but the effect on
the Gabriel-Roiter theory was not yet known.