The root posets and their rich antichains

TL;DR

This paper studies the structure of root posets associated with Dynkin diagrams, revealing their conical nature, properties of rich antichains, and the existence of a unique maximal rich antichain with specific length characteristics.

Contribution

It demonstrates that root posets are conical, characterizes rich antichains, and establishes the existence and uniqueness of a maximal rich antichain with roots of uniform length, except for E6.

Findings

Root posets are disjoint unions of n chains.

Rich antichains have cardinality n-1 and are equinumerous with positive roots.

A unique rich antichain contains all other rich antichains within its generated ideal.

Abstract

Let $\Delta$ be a (connected) Dynkin diagram of rank $n\ge 2$ and $\Phi_+ = \Phi_+(\Delta)$ the corresponding root poset (it consists of all positive roots with respect to a fixed root basis). The width of $\Phi_+$ is $n$. We will show that $\Phi_+$ is "conical": it is the disjoint union of $n$ solid chains. The rich antichains in $\Phi_+$ are the antichains of cardinality $n-1$. It is well known that the number of rich antichains is equal to the cardinality of $\Phi_+$. The set $\mathcal R(\Delta)$ of rich antichains in $\Phi_+$ can itself be considered as a poset which is quite similar, but not always isomorphic, to $\Phi_+$. We will show that there always exists a unique rich antichain $A$ such that any rich antichain is contained in the ideal generated by $A$. For $\Delta\neq \Bbb E_6$ all roots in $A$ have the same length, namely $e_2$, where $e_1 \le e_2 \le \dots \le e_n$ are the exponents of $\Delta.$ For $\Delta = \Bbb E_6$, the antichain $A$ consists of four roots of length $e_2 = 4$ and one root of length $5$.