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Transitions between root subsets associated with Carter diagrams

Rafael Stekolshchik

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 $Δ(E_6)$, $Δ(E_7)$ and $Δ(E_8)$ (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 $E_6$, $E_7$, $E_8$.

Transitions between root subsets associated with Carter diagrams

Abstract

For any two root subsets associated with two Carter diagrams that have the same 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 , and (introduced by Dynkin-Minchenko) and Carter diagrams. This relationship echoes the assertions obtained by Ringel, Rosenfeld and Baez in completely different contexts regarding the Dynkin diagrams , , .
Paper Structure (39 sections, 47 equations, 10 figures, 3 tables)

This paper contains 39 sections, 47 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Diagrams with cycles
  3. Homogeneous classes of Carter diagrams
  4. A group of transitions $\mathscr{M}$
  5. Theorems on transition matrix
  6. Relationship with other diagrams containing cycles
  7. Dynkin-Minchenko diagrams: Procedure of completion
  8. Values 2,4,8 and diagrams $E_6$, $E_7$, $E_8$
  9. McKee-Smyth diagrams: Eigenvalues in $(-2,2)$
  10. Diagrams containing cycles
  11. Admissible diagrams: Conjugacy classes of $W$
  12. Carter diagrams: Language of "solid and dotted" edges
  13. Carter diagrams: Eliminating the cycle.
  14. Connection diagrams
  15. Bicolored partition
  16. ...and 24 more sections

Figures (10)

  • Figure 1: Carter diagrams of $D$ and $E$ types
  • Figure 2: Diagram $\Gamma_1$ (resp. $\Gamma_2$) of type $D_4(a_1)$ (resp. equivalent to $D_4$)
  • Figure 3: Conjugate elements $\{ w_o , w_{\widetilde{o}}\}$ corresponding to the Coxeter class $D_4$
  • Figure 4: Eight similar $4$-cycles equivalent to $D_4(a_1)$)
  • Figure 5: Alternative transitions $\{E_8(a_6), E_8(a_5) \}$ and $\{E_8(a_6), E_8(a_1) \}$)
  • ...and 5 more figures

Theorems & Definitions (4)

  • proof
  • proof
  • proof
  • proof