Integer quadratic forms and extensions of subsets of linearly independent roots

TL;DR

This paper investigates the conditions under which roots can be added to linearly independent root subsets in root systems, introducing linkage roots and systems, and analyzing their structure and group actions.

Contribution

It establishes a criterion for linkage roots using quadratic forms and explores the structure and orbits of linkage systems associated with Carter diagrams.

Findings

Linkage roots satisfy a quadratic form inequality involving the inverse Cartan matrix.

The size of linkage systems is invariant under conjugation of Cartan matrices.

The structure and orbits of linkage systems for specific Dynkin diagrams are characterized.

Abstract

We consider subsets of linearly independent roots in a certain root system $\varPhi$. Let $S'$ be such a subset, and let $S'$ be associated with any Carter diagram $\Gamma'$. The main question of the paper: what root $\gamma \in \varPhi$ can be added to $S'$ so that $S' \cup \gamma$ is also a subset of linearly independent roots? This extra root $\gamma$ is called the linkage root. The vector $\gamma^{\nabla}$ of inner products $\{(\gamma,\tau'_i)\mid \tau'_i \in S'\}$ is called the linkage label vector. Let $B_{\Gamma'}$ be the Cartan matrix associated with $\Gamma'$. It is shown that $\gamma$ is a linkage root if and only if $\mathscr{B}^{\vee}_{\Gamma'}(\gamma^{\nabla}) < 2$, where $\mathscr{B}^{\vee}_{\Gamma'}$ is a quadratic form with the matrix inverse to $B_{\Gamma'}$. The set of all linkage roots for $\Gamma'$ is called a linkage system and is denoted by $\mathscr{L}(\Gamma')$. The Cartan matrix associated with any Carter diagram $\Gamma'$ is conjugate to the Cartan matrix associated with some Dynkin diagram $\Gamma$, [St23]. The sizes of $\mathscr{L}(\Gamma')$ and $\mathscr{L}(\Gamma)$ are the same. Let $W^{\vee}$ be the Weyl group of the quadratic form $\mathscr{B}^{\vee}_{\Gamma'}$. This group acts on the linkage system and forms several orbits. The sizes and structure of orbits for linkage systems $\mathscr{L}(D_l)$ and $\mathscr{L}(D_l(a_k))$ are presented.