Kleiner's theorem for unitary representations of posets

TL;DR

This paper extends Kleiner's theorem to unitary representations of posets, characterizing when there are finitely many indecomposable representations based on the structure of the poset.

Contribution

It introduces the concept of unitary hi-representations and establishes a criterion linking their finiteness to the classical subspace representation finiteness for posets.

Findings

Finite hi-representations occur only for posets with finitely many subspace representations.

The classification hinges on the presence of Kleiner's critical posets.

The result generalizes Kleiner's theorem to the setting of unitary representations.

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 $\chi_1 P_1+...+\chi_t P_t= \mathbb 1$, 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.