Mikhail V. Ignatev ; Matvey A. Surkov - Rook placements in $G_2$ and $F_4$ and associated coadjoint orbits

cm:9041 - Communications in Mathematics, March 22, 2023, Volume 30 (2022), Issue 2 (Special Issue: CIMPA School "Nonassociative Algebras and Its Applications", Madagascar 2021) - https://doi.org/10.46298/cm.9041
Rook placements in $G_2$ and $F_4$ and associated coadjoint orbitsArticle

Authors: Mikhail V. Ignatev ; Matvey A. Surkov

    Let $\mathfrak{n}$ be a maximal nilpotent subalgebra of a simple complex Lie algebra with root system $\Phi$. A subset $D$ of the set $\Phi^+$ of positive roots is called a rook placement if it consists of roots with pairwise non-positive scalar products. To each rook placement $D$ and each map $\xi$ from $D$ to the set $\mathbb{C}^{\times}$ of nonzero complex numbers one can naturally assign the coadjoint orbit $\Omega_{D,\xi}$ in the dual space $\mathfrak{n}^*$. By definition, $\Omega_{D,\xi}$ is the orbit of $f_{D,\xi}$, where $f_{D,\xi}$ is the sum of root covectors $e_{\alpha}^*$ multiplied by $\xi(\alpha)$, $\alpha\in D$. (In fact, almost all coadjoint orbits studied at the moment have such a form for certain $D$ and $\xi$.) It follows from the results of Andrè that if $\xi_1$ and $\xi_2$ are distinct maps from $D$ to $\mathbb{C}^{\times}$ then $\Omega_{D,\xi_1}$ and $\Omega_{D,\xi_2}$ do not coincide for classical root systems $\Phi$. We prove that this is true if $\Phi$ is of type $G_2$, or if $\Phi$ is of type $F_4$ and $D$ is orthogonal.


    Volume: Volume 30 (2022), Issue 2 (Special Issue: CIMPA School "Nonassociative Algebras and Its Applications", Madagascar 2021)
    Published on: March 22, 2023
    Accepted on: February 3, 2022
    Submitted on: February 3, 2022
    Keywords: Mathematics - Representation Theory,17B08, 17B10, 17B22, 17B25, 17B30, 17B35

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