Weyl group actions

Abstract

One main purpose of the Local Structure Theorem for full heaps (Theorem 2.3.15) is to enable the construction of various familiar algebraic objects using full heaps as a starting point. We explain in Section 3.1 how to associate linear operators with full heaps; these linear operators will act on vector spaces whose bases are indexed by ideals of the heap itself. The most important linear operators for our purposes will be the operators X p , Y p , H p , S p and T p, q , all of which are defined in Section 3.1. In Chapter 3, we will concentrate on the operators S p and T p,q , which will give rise to representations of the associated Weyl group (Theorem 3.2.27) and Hecke algebra (Theorem 3.1.13) respectively. The operators X p , Y p and H p will reappear later in the construction of Lie algebras. It turns out to be more convenient to consider the action of the linear operators on a subset of the ideals of the full heap. These are the so-called “proper ideals”: an ideal is proper if its intersection with each vertex chain of the original heap is both proper and nonempty, considered as a set. Section 3.2 develops the theory of proper ideals, and also introduces the key concepts of the “content” of a finite heap and the “relative content” of a pair of proper ideals. The action of the Weyl group on the root system emerges naturally from these concepts and gives rise to Theorem 3.2.21, which gives a precise connection between the action of the Weyl group on the proper ideals and the action on the root system.