Prenilpotent pairs in the E10 root system

TL;DR

This paper investigates the complexity of prenilpotent pairs in the E10 root system, revealing an exponential growth in their orbits, which challenges direct enumeration and suggests the need for alternative methods in studying Kac-Moody groups.

Contribution

It demonstrates the rapid growth of prenilpotent pair orbits in E10, highlighting the impracticality of explicit enumeration and motivating new approaches to understanding Tits' groups.

Findings

Number of prenilpotent pair orbits grows at least as fast as a constant times the seventh power of the inner product k.

Explicit enumeration of all orbits in E10 is infeasible due to their rapid growth.

Results motivate the development of alternative methods for studying Kac-Moody groups.

Abstract

Tits 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)(7th power of k). Our purpose is to motivate alternate approaches to Tits' groups.