Prenilpotent pairs in the E10 root lattice

Abstract

AbstractTits has defined Kac–Moody groups for all root systems, over all commutative rings with unit. A central concept is the idea of a prenilpotent pair of (real) roots. In particular, writing down his group presentation explicitly would require knowing all the Weyl-group orbits of such pairs. We show that for the hyperbolic root system E10 there are so many orbits that any attempt at direct enumeration is impractical. Namely, the number of orbits of prenilpotent pairs having inner product k grows at least as fast as (constant) ⋅ k7 as k → ∞. Our purpose is to motivate alternate approaches to Tits' groups, such as the one in [2].