Representation types of the category of subprojective representations of a finite poset over K[t]/(tm) and a solution of a Birkhoff type problem

We determine the representation type (wild, tame, polynomial growth) of the category fspr(I,Fm) of filtered subprojective Fm-representations of a finite poset I in terms of m and I, where Fm=K[t]/(tm), m⩾1, and K is an algebraically closed field. Criteria for tameness, wildness and tameness of non-polynomial growth of fspr(I,Fm) are given in Theorems 1.1 and 1.2. As an application, a solution of Birkhoff's type problem [G. Birkhoff, Subgroups of abelian groups, Proc. London Math. Soc. 38 (1934) 385–401] for the category repft(I,Fm) of filtered I-chains of Fm-modules is given in Section 5, by determining the representation type repft(I,Fm).