8 - Combinatorics of Weyl groups

Summary

The weights of a minuscule representation may be regarded as points in Euclidean space. As we shall see in Chapter 8, the convex hull of these points forms a polytope with interesting combinatorial properties, and the action of the Weyl group on the polytope gives additional insight into the nature of minuscule representations.

Section 8.1 introduces the notion of a minuscule system. This provides a convenient way to describe explicit coordinates for the weights of all minuscule representations. This is useful for later purposes when concrete constructions are required.

Section 8.2 describes the action of the Weyl group as a permutation group on the weights of a minuscule representation. It is well-known that this action is transitive, but we go further and describe the W -orbits on ordered pairs of weights. This turns out to be important for some later combinatorial constructions.

Section 8.3 describes the remarkable relationship between the weights of a minuscule representation of weight ω_p and the positive roots of the Weyl group in which α_p appears with nonzero coefficient.

Section 8.4 introduces the weight polytopes of minuscule representations; that is, the convex hull of the set of weights of a minuscule representation. Section 8.5 analyses the combinatorics of the faces of the weight polytope. Finally, Section 8.6 shows how to associate families of graphs with the weight polytopes. These graphs come equipped with an action of the Weyl group, and include several families of graphs that are of independent interest.