FULL HEAPS AND REPRESENTATIONS OF AFFINE WEYL GROUPS

Year 2008, Volume: 3 Issue: 3, 1 - 42, 01.06.2008

R. M. Green

Abstract

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 E6 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.

Keywords

affine Weyl groups, heaps of pieces, 27 lines on a cubic surface, permutation representations, root systems, Kac-Moody algebras

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• Department of Mathematics, University of Colorado Campus Box 395 Boulder, CO 80309-0395 USA
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