Comparison of Auslander–Reiten Theory and Gabriel–Roiter Measure Approach to the Module Categories of Tame Hereditary Algebras

Abstract

Let Λ be a tame hereditary algebra over an algebraically closed field, i.e., Λ = kQ with Q a quiver of type , , , , or . Two different kinds of partitions of the module category can be obtained by using Auslander–Reiten theory, and on the other hand, Gabriel–Roiter measure approach. We compare these two kinds of partitions and see how the modules are rearranged according to Gabriel–Roiter measure. We also show that the Gabriel–Roiter submodules can be used to build orthogonal exceptional pairs for indecomposable preprojective Λ-modules when Λ is of type and .