On the Representation Theory of Incidence Algebras

In this thesis, we will discuss the representation theory of incidence algebras, a particular class of algebras arising from partially ordered sets. Given any algebra of finite representation type, it is known that it is also tau-tilting finite, but the converse is not true in general. The first part of this work shows that there is an equivalence between representation finiteness and tau-tilting finiteness for incidence algebras, using the theory of concealed algebras and tau-tilting theory. In the last part, the focus shifts to the study of generic bricks over the representation-infinite incidence algebras. Generic bricks are generic modules, i.e. modules with infinite length but finite endolength, that are also bricks. Using the notions of brick-continuous algebra, Tits quadratic form of a quiver and reflection functors, a method is given to explicitly construct and compute a generic brick for any incidence algebra of infinite representation type.