Weyl group orbits on Kac--Moody root systems

TL;DR

This paper characterizes the orbit structure of Weyl groups acting on roots in Kac--Moody algebras, revealing conditions for orbit equivalence, and analyzing the impact of matrix zeros and hyperbolic geometry on root orbits.

Contribution

It provides a complete characterization of Weyl group orbits on roots, introduces the Cayley graph for orbit analysis, and explores orbit properties in hyperbolic Kac--Moody algebras.

Findings

Simple roots are in the same orbit iff their vertices are connected by a path of single edges.

In symmetric hyperbolic cases, roots of the same length lie in the same orbit.

Presence of zeros in the Cartan matrix leads to non-simple transitivity of the Weyl group action.

Abstract

Let $\mathcal{D}$ be a Dynkin diagram and let $\Pi=\{\alpha_1,\dots ,\alpha_{\ell}\}$ be the simple roots of the corresponding Kac--Moody root system. Let $\mathfrak{h}$ denote the Cartan subalgebra, let $W$ denote the Weyl group and let $\Delta$ denote the set of all roots. The action of $W$ on $\mathfrak{h}$, and hence on $\Delta$, is the discretization of the action of the Kac--Moody algebra. Understanding the orbit structure of $W$ on $\Delta$ is crucial for many physical applications. We show that for $i\neq j$, the simple roots $\alpha_i$ and $\alpha_j$ are in the same $W$--orbit if and only if vertices $i$ and $j$ in the Dynkin diagram corresponding to $\alpha_i$ and $\alpha_j$ are connected by a path consisting only of single edges. We introduce the notion of `the Cayley graph $\mathcal{P}$ of the Weyl group action on real roots' whose connected components are in one-to-one correspondence with the disjoint orbits of $W$. For a symmetric hyperbolic generalized Cartan matrix $A$ of rank $\geq 4$ we prove that any 2 real roots of the same length lie in the same $W$--orbit. We show that if the generalized Cartan matrix $A$ contains zeros, then there are simple roots that are stabilized by simple root reflections in $W$, that is, $W$ does not act simply transitively on real roots. We give sufficient conditions in terms of the generalized Cartan matrix $A$ (equivalently ${\mathcal D}$) for $W$ to stabilize a real root. Using symmetry properties of the imaginary light cone in the hyperbolic case, we deduce that the number of $W$--orbits on imaginary roots on a hyperboloid of fixed radius is bounded above by the number of root lattice points on the hyperboloid that intersect the closure of the fundamental region for $W$.