A geometric realization of socle-projective categories for posets of type [formula omitted

Abstract

This paper establishes a link between the theory of cluster algebras and the theory of representations of partially ordered sets. We introduce a class of posets by requiring avoidance of certain types of peak-subposets and show that these posets can be realized as the posets of quivers of type A with certain additional arrows. This class of posets is therefore called posets of typeA. We then give a geometric realization of the category of finitely generated socle-projective modules over the incidence algebra of a poset of type A as a combinatorial category of certain diagonals of a regular polygon. This construction is inspired by the realization of the cluster category of type A as the category of all diagonals by Caldero, Chapoton and the first author [10]. We also study the subalgebra of the cluster algebra generated by those cluster variables that correspond to the socle-projectives under the above construction. We give a sufficient condition for when this subalgebra is equal to the whole cluster algebra.