On a criterion of the finiteness of the representation type for families of the categories of injective representations

Abstract

The representations of posets (partially ordered sets), introduced by L. A. Nazarova and A. V. Roiter in 1972, play an important role in the modern representation theory and its applications. M. M. Kleiner obtained a description of posets of finite representation type in terms of critical posets (the minimal ones of infinite representation type) and Yu. A. Drozd proved that a poset S (not containing an element designated as 0) is of finite representation type if and only if its Tits quadratic form is weakly positive, i.e. positive on the set of non-negative vectors (in 1972 and 1974, respectively). In this paper we consider a situation (which deals with infinite posets), when the main role is played not by weakly positivity but by positivity of the Tits quadratic form. The situation relates to the study of the categories of representations of a special form, and in this case we use established by the first author a connection between the Tits quadratic forms for partially ordered sets and commutative quivers.

Keywords

• injective representation
• critical poset
• tits quadratic form for commutative quivers
• finite representation type
• positivity and weak positivity