Coxeter spectral classification of non-negative posets of Dynkin type \(\mathbb{E}_m\)

Abstract

We give a complete description of connected non-negative Dynkin type Dyn\(_I=\mathbb{E}_m\) posets and prove that the number of such posets is finite. Moreover, by means of computer assisted analysis, we give a complete Coxeter classification of this class and prove that the pair (Dyn\(_I=\mathbb{E}_m,\) specc\(_I\)), where specc\(_I\subseteq\mathbb{C}\) denotes the Coxeter spectrum of \(I\), determines \(I\) uniquely, up to the strong Gram \(\mathbb{Z}\)-congruence.