Posets of finite prinjective type and a class of orders

Abstract

The main aim of this paper is to give a characterisation of finite posets J having only finitely many isomorphism classes of indecomposable socle projective K-linear representations over a given field K, or equivalently, finite posets J having only finitely many indecomposable canonical forms of partitioned matrices of the shape (2.12) (with coefficients in K) with respect to the J-elementary transformations (E 1) and (E 2) defined in Section 2. The characterisation is given in Theorem 3.1 in terms of the Tits quadratic form associated to J, in terms of a class of algebraic varieties with an algebraic group action, and by presenting a critical list of 114 minimal posets having infinitely many isomorphism classes of indecomposable socle projective representations. An application of posets of finite prinjective type to the study of indecomposable lattices over a class of orders is given.