Abstract: The aim of this lecure will be to review two different approaches to the use of the representation theory of hereditary artin algebras A for dealing with the root poset of a finite root system, following on the one hand Ingalls-Thomas (2009), on the other hand Dlab-Ringel (1979). We will focus the attention to two kinds of antichains of positive roots: there are the antichains in the root poset itself with its usual partial ordering (this ordering corresponds to the subfactor ordering of indecomposable A-modules), the second kind depends on the choice of a Coxeter element: here we deal with antichains in the category of A-modules, or, equivalently, with generalized non-crossing partitions.