Kleiner’s theorem for unitary representations of posets

A subspace representation of a poset S={s1,…,st} is given by a system (V;V1,…,Vt) consisting of a vector space V and its subspaces Vi such that Vi⊆Vj if si≺sj. For each real-valued vector χ=(χ1,…,χt) with positive components, we define a unitary χ-representation of S as a system (U;U1,…,Ut) that consists of a unitary space U and its subspaces Ui such that Ui⊆Uj if si≺sj and satisfies χ1P1+⋯+χtPt=1, in which Pi is the orthogonal projection onto Ui.We prove that S has a finite number of unitarily nonequivalent indecomposable χ-representations for each weight χ if and only if S has a finite number of nonequivalent indecomposable subspace representations; that is, if and only if S contains any of Kleiner’s critical posets.