Incidence algebras of posets, or structural matrix algebras, have been studied from various perspectives, including combinatorics, linear algebra, representation theory. We introduce a deformation theory for such algebras, in terms of the simplicial realization of the underlying poset, and give a unified approach to understanding several classes of representations (such as with finitely many orbits, with finitely many invariant subspaces, or which are distributive, or which are thin, i.e. the multiplicity of each simple in the composition series is less than 1). We and give a series of applications.

First, we give new characterizations of incidence algebras (as algebras with a faithful distributive representation) and show that their deformations are exactly the locally hereditary algebras which are semi-distributive or have finitely many ideals. Second, we give a complete classification of all thin and all distributive representations of incidence algebras. Furthermore, we extend this to arbitrary algebras and provide a method to completely classify thin representations over any algebra. As a consequence, we obtain the following ``generic classification": we show that any thin representation of any algebra, and any distributive representation of an acyclic algebra can be presented, by choosing suitable bases and after canceling the annihilator, as the defining representation of an incidence algebra. Time permitting, we discuss other applications, such as consequences on the structure of representation and Grothendieck rings of incidence algebras, and an answer to a conjecture of Bongartz and Ringel, in a particular case.