Some remarks on Hilbert representations of posets

TL;DR

This paper investigates Hilbert space representations of certain finite posets, establishing conditions for finite-dimensionality and characterizing when essential irreducible orthoscalar representations exist.

Contribution

It provides a classification of finite-dimensional irreducible orthoscalar representations for a specific class of finite posets and identifies when essential representations occur.

Findings

All irreducible orthoscalar representations are finite-dimensional for the studied posets.

Criteria are given for the existence of essential irreducible orthoscalar representations.

The paper characterizes which posets admit non-degenerate irreducible orthoscalar representations.

Abstract

For a certain class of finite posets, we prove that all their irreducible orthoscalar representations are finite-dimensional and describe those, for which there exist essential (non-degenerate) irreducible orthoscalar representations.