Kleiner's theorem for unitary representations of posets

A subspace representation of a poset $\mathcal{S}=\{s_1,\mathrm{.}\mathrm{.}\mathrm{.},s_t\}$ is given by a system $(V;V_1,\mathrm{.}\mathrm{.}\mathrm{.},V_t)$ consisting of a vector space $V$ and its subspaces $V_i$ such that $V_i\subseteq V_j$ if $s_i\prec s_j$. For each real-valued vector $\chi =({\chi }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\chi }_t)$ with positive components, we define a unitary $\chi$-representation of $\mathcal{S}$ as a system $(U;U_1,\mathrm{.}\mathrm{.}\mathrm{.},U_t)$ that consists of a unitary space $U$ and its subspaces $U_i$ such that $U_i\subseteq U_j$if$s_i\prec s_j$andsatisfies$\chi_1 P_1+…+\chi_t P_t= \mathbb 1$, in which $P_i$ is the orthogonal projection onto $U_i$. We prove that $\mathcal{S}$ has a finite number of unitarily nonequivalent indecomposable $\chi$-representations for each weight $\chi$ if and only if $\mathcal{S}$ has a finite number of nonequivalent indecomposable subspace representations; that is, if and only if $\mathcal{S}$ contains any of Kleiner’s critical posets.