A description of the outer automorphism of S6, and the invariants of six points in projective space

We use a simple description of the outer automorphism of S6 to cleanly describe the invariant theory of six points in P, P, and P. In §1.1–1.2, we give two short descriptions of the outer automorphism of S6, complete with a proof that they indeed describe an outer automorphism. (Our goal is to have a construction that the reader can fully understand and readily verify.) In §1.3 we give another variation on this theme that is attractive, but longer. The latter two descriptions do not distinguish any of the six points. Of course, these descriptions are equivalent to the traditional one (§1.4) — there is after all only one nontrivial outer automorphism (modulo inner automorphisms). In §2, we use this to cleanly describe the invariant theory of six points in projective space. This is not just a random application; the descriptions of §1 were discovered by means of this invariant theory. En route we use the outer automorphism to describe five-dimensional representations of S5 and S6, §1.5. The outer automorphism was first described by Hölder in 1895. Most verifications use some variation of Sylvester’s synthemes, or work directly with generators of S6; nontrivial calculation is often necessary. Other interpretations are in terms of finite geometries, for example involving finite fields with 2, 3, 4, 5, or 9 elements, and are beautiful, but require non-trivial verification. 1. THE OUTER AUTOMORPHISM OF S6 1.1. First description of the outer automorphism: the mystic pentagons. Consider a complete graph on five vertices numbered 1 through 5. The reader will quickly verify that there are precisely six ways to two-color the edges (up to choices of colors) so that the edges of one color (and hence the other color) form a 5-cycle, see Figure 1. We dub these the six mystic pentagons. Then S5 acts on the six mystic pentagons by permuting the vertices, giving a map i : S5 = S{1,...,5} → S{a,...,f} = S6. This is an inclusion — the kernel must be one of the normal subgroups {e}, A5, or S5, but we visually verify that (123) acts nontrivially. Moreover, it is not a usual inclusion as (12) induces permutation (ad)(bc)(ef) — not a transposition. Hence S6 = S{a,...,f} acts on the six cosets of i(S5), inducing a map f : S{a,...,f} → S{1,...,6}. This is the outer automorphism. This can be verified in several ways (e.g., (ad)(bc)(ef) induces the nontrivial permutation (12) ∈ Date: January 23, 2008. 1991 Mathematics Subject Classification. Primary 20B30, Secondary 20C30, 14L24. Partially supported by NSF CAREER/PECASE Grant DMS–0228011. 1 S{1,...,6}, so f is injective and hence an isomorphism; and i is not a usual inclusion, so f is not inner), but for the sake of simplicity we do so by way of a second description.