Rafael Stekolshchik - Transitions between root subsets associated with Carter diagrams

cm:9760 - Communications in Mathematics, July 11, 2023, Volume 30 (2022), Issue 3 (Special issue: in memory of Arkady Onishchik) - https://doi.org/10.46298/cm.9760
Transitions between root subsets associated with Carter diagramsArticle

Authors: Rafael Stekolshchik

    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 $E_6$, $E_7$, $E_8$.


    Volume: Volume 30 (2022), Issue 3 (Special issue: in memory of Arkady Onishchik)
    Published on: July 11, 2023
    Accepted on: June 6, 2023
    Submitted on: July 5, 2022
    Keywords: Mathematics - Representation Theory,20F55, 20E45, 15A18

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