Transitions between root subsets associated with Carter diagrams

TL;DR

This paper studies the relationships between root subsets linked to Carter diagrams of the same type and size, constructing involutive transition matrices that demonstrate their conjugacy under the Weyl group, and explores numerical patterns connecting Dynkin and Carter diagrams.

Contribution

It introduces a canonical involutive transition process between root subsets associated with Carter diagrams of the same type, revealing their conjugacy under the Weyl group and uncovering numerical relationships with enhanced Dynkin diagrams.

Findings

All root subsets associated with a Carter diagram are conjugate under the Weyl group.

Constructed transition matrices are involutions that map one root subset to another.

Numerical relationships connect enhanced Dynkin diagrams with Carter diagrams, echoing classical $2-4-8$ patterns.

Abstract

For any two root subsets associated with two Carter diagrams that have the same $ADE$ type and the same size, we construct the transition matrix that maps one subset to the other. The transition between these two subsets is carried out in some canonical way affecting exactly one root, so that this root is mapped to the minimal element in some root subsystem. The constructed transitions are involutions. It is shown that all root subsets associated with the given Carter diagram are conjugate under the action of the Weyl group. A numerical relationship is observed between enhanced Dynkin diagrams $\Delta(E_6)$, $\Delta(E_7)$ and $\Delta(E_8)$ (introduced by Dynkin-Minchenko) and Carter diagrams. This relationship echoes the $2-4-8$ assertions obtained by Ringel, Rosenfeld and Baez in completely different contexts regarding the Dynkin diagrams $E_6$, $E_7$, $E_8$.