Title:

On a triangulated category which models positive noncrossing partitions

Let Q be a simply laced Dynkin quiver, Db(Q) the bounded derived category of the path
algebra associated to Q and C(Q) the TΣ2orbit category ofDb( Q), where T is the Auslander
Reitentranslation and Σ is the shift functor. Note that C( Q) is a triangulated category [57].
In this thesis we study two classes of representationtheoretic objects in C ( Q): maximal Hom
free objects, also known as Hornconfigurations, and maximal rigid (i.e. Extfree) objects.
Hornconfigurations (in the derived category) were used by Riedtrnann [65] in order to
classify selfinjective algebras of finiterepresentation type. Riedtmann proved that these
objects are invariant under the autoequivalence TΣ2 of Db(Q). This is the reason why we
consider the orbit category C( Q).
We establish a bijection between Hornconfigurations and noncrossing partitions of the
Coxeter group associated to Q which are not contained in any proper standard parabolic
subgroup. These noncrossing partitions are said to be positive, because they are proved to
be in 11 correspondence with the positive clusters in the corresponding cluster algebra (cf.
[62]).
The bijection between Hornconfigurations and positive noncrossing partitions generalizes a
result of Riedtmann which states that Hornconfigurations in Db (Q), where Q is of type An'
are in bijection with classical noncrossing partitions of the set {1, ... , n}.
Riedtmann's bijection allows us to construct a geometrical model for C(Q) in type A. Using
this geometrical setup, and inspired by the classification of the clustertilting objects in the
cluster category of type An in terms of triangulations of a polygon, we classify, also in type
An, the maximal rigid objects in C(Q) in terms of certain noncrossing bipartite graphs. In
addition, we describe the corresponding endomorphism algebras in terms of quivers with
relations.
We also give a natural notion of mutation of Hornconfigurations in C(Q) and we present
some partial results and conjectures about the mutation graph and the representationtheoretic
description of these mutations
