6 Full Heaps and Representations of Affine Weyl Groups

We use the author's combinatorial theory of full heaps to categorify the action of a large class of Weyl groups on their root systems, and thus to give an elementary and uniform construction of a family of faithful permutation representations of Weyl groups. Examples include the standard representations of affine Weyl groups as permutations of Z and geometrical examples such as the realization of the Weyl group of type E 6 as permutations of 27 lines on a cubic surface; in the latter case, we also show how to recover the incidence relations between the lines from the structure of the heap. Another class of examples involves the action of certain Weyl groups on sets of pairs (t, f), where t ∈ Z and f is a function from a suitably chosen set to the two-element set {+, −}. Each of the permutation representations corresponds to a module for a Kac–Moody algebra, and gives an explicit basis for it.