Weyl group orbits on Kac-Moody root systems

• Published in: Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2014-11, Vol.47 (44), p.445201-25
• Main Authors: Carbone, Lisa, Conway, Alexander, Freyn, Walter, Penta, Diego
• Format: Article
• Language: English
• Subjects: Closures; Kac-Moody algebra; Lattices; Mathematical analysis; Orbits; root system; Roots; Symmetry; Texts; Weyl group orbits
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8113/47/44/445201

Let be a Dynkin diagram and let be the simple roots of the corresponding Kac-Moody root system. Let denote the Cartan subalgebra, let W denote the Weyl group and let denote the set of all roots. The action of W on , and hence on , is the discretization of the action of the Kac-Moody algebra. Understanding the orbit structure of W on is crucial for many physical applications. We show that for , the simple roots i and j are in the same W-orbit if and only if vertices i and j in the Dynkin diagram corresponding to i and j are connected by a path consisting only of single edges. We introduce the notion of 'the Cayley graph 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 we prove that any two 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 ) 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.