Schur's Lemma for partially ordered sets of finite type

Abstract

It follows, for example, from the explicit description of the indecomposable representations of posets of finite type (51. Being interesting per se, Schur’s Lemma easily implies a bijection between the dimensions of indecomposable representations of a poset of finite type and the positive integral roots of its Tits form. That was shown in [ 1 ] according to the idea of [2] where the same implication was established for representations of quivers of finite type. This bijection allows one to prove directly certain qualitative results that otherwise follow only from the classification of representations. There are two published proofs of Schur’s Lemma for posets. One of them has a gap [7], and the other [I] is rather cumbersome. The purpose of this paper is to give a simple proof of the Lemma. Together with the well-known fact that one can canonically associate an indecomposable representation of a poset of finite type to every indecomposable representation of a quiver of finite type [2], the proof below constitutes perhaps the shortest known proof of Schur’s Lemma for quivers of finite type as well.