Representations of finite posets over the ring of integers modulo a prime power

Abstract

The classical category Rep$(S,\mathbb {Z}_{p})$ of representations of a finite poset $S$ over the field $\mathbb {Z}_{p}$ is extended to two categories, Rep$(S,\mathbb {Z}_{p^{m}})$ and uRep$(S,\mathbb {Z}_{p^{m}})$, of representations of $S$ over the ring $\mathbb {Z}_{p^{m}}$. A list of values of $S$ and $m$ for which Rep$(S,\mathbb {Z}_{p^{m}})$ or uRep$(S,\mathbb {Z}_{p^{m}})$ has infinite representation type is given for the case that $S$ is a forest. Applications include a computation of the representation type for certain classes of abelian groups, as the category of sincere representations in (uRep$(S,\mathbb {Z}_{p^{m}})$) Rep$(S,Z_{p^{m}})$ has the same representation type as (homocyclic) $(S,p^{m})$-groups, a class of almost completely decomposable groups of finite rank. On the other hand, numerous known lists of examples of indecomposable $(S,p^{m})$-groups give rise to lists of indecomposable representations.