10 - Exceptional structures

Summary

It follows from Example 8.2.11 and Exercise 8.2.12 that there is a tower of vector spaces

L(A₅, ω₃) ⊂ L(D₆, ω₆) ⊂ L(E₇, ω₆)

of dimensions 20, 32 and 56. This inclusion respects the decomposition of each space into weight spaces. This corresponds to a chain

W(A₅)/W(A₂ ∪ A₂) ⊂ W(D₆)/W(A₅) ⊂ W(E₇)/W(E₆)

of containments of cosets of parabolic subgroups. (In the case of W(D₆), we may use either of the two parabolic subgroups of type A₅.) This chapter is about the surprisingly rich combinatorial structure of this chain of inclusions.

Section 10.1 introduces the famous configuration of 27 lines on a cubic surface. The symmetry group of this configuration is well-known to be the group W(E₆). We shall see that the incidence properties of these lines turn out to be governed by the combinatorics of the minuscule module L(E₆, ω₅). We develop these ideas further in Section 10.2, which investigates the properties of Schläfli double sixes; these are particular collections of 12 lines on the cubic surface.

Section 10.3 introduces the theory of 2-graphs. Just as a graph may be regarded as a collection of distinguished pairs (i.e., edges) in a vertex set, a 2-graph may be regarded as a collection of distinguished triples in a vertex set. We shall see that there is an interesting 2-graph on the set of 28 bitangents in which the distinguished triples are the azygetic triples.