Kleiner's theorem for unitary representations of posets

Abstract

A subspace representation of a poset $\mathcal{S}=\{s_1,...,s_t\}$ is given by a system $(V;V_1,...,V_t)$ 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 $\chi =({\chi }_1,...,{\chi }_t)$ with positive components, we define a unitary $\chi$-representation of $\mathcal{S}$ as a system $(U;U_1,...,U_t)$ that consists of a unitary space $U$ and its subspaces $U_i$ such that $U_i\subseteq U_j$ if $s_i\prec s_j$ and satisfies ${\chi }_1{P}_1+...+{\chi }_t{P}_t=1$, 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.

12 pages, paper reorganized and rewritten. some statements were added

References in corpus (1)

• Stable Configurations of Linear Subspaces and Quotient Coherent Sheaves