seam

Floor 2, Les sutures · introduced in Chapter 1, Un objet, plusieurs noms

How are two incarnations of one object matched?

An equivariant bijection between two incarnations of one object; the seams between two incarnations form a torsor under the object’s automorphisms.

the Sylow subgroup fixing both

{0, ∞}⟨z ↦ 2z⟩(1, 246)

1 of 28

Plate 2.1The seam, for one marking, from the pairs of

\Proj^1(\F_7)

to the antiflags of the Fano plane, element by element, with the Sylow 3-subgroup that fixes both. It carries neighbours to neighbours.

Definition(Seam)

A seam between two incarnations $Y$ and $Y'$ of an object is a $G$ -isomorphism $Y\to Y'$ . Seams compose, and the inverse of a seam is a seam.

A seam is the line along which two pieces of cloth are joined. Here the pieces are theories, and the join is an equivariant bijection between the sets in which they meet one object.

Theorem(The stabilizer principle)

Let $X$ be an object with stabilizer class $[H]$ . Then $\Aut_G(X)\cong N_G(H)/H$ , and for two incarnations $Y$ and $Y'$ the seams $Y\to Y'$ form a torsor for $\Aut_G(Y)$ acting by precomposition and for $\Aut_G(Y')$ acting by postcomposition. There are exactly $|N_G(H):H|$ of them, and one is written down as soon as a pair of points with equal stabilizers is found: $gy\mapsto gy'$ .

Proof

If $s,s'$ are seams, $s^{-1}s'$ is an automorphism of $Y$ and $s'=s\circ(s^{-1}s')$ . The automorphisms of $X=Gx$ correspond to the points $nx$ with $n\in N_G(H)$ , and $nx=x$ exactly when $n\in H$ .

Proposition(The seams, explicitly)

Let $Y$ be any incarnation of the object of size 28. Each Sylow 3-subgroup $P$ of $G$ fixes exactly one point $y_P$ of $Y$ , the point with stabilizer $N_G(P)$ , and $P\mapsto y_P$ is the unique seam from the Sylow subgroups to $Y$ .

On $\Proj^1(\F_7)$ , $y_P$ is the pair of points fixed by $P$ ; in the Fano plane, the unique point and the unique line fixed by $\mu_A(P)$ ; on the Klein quartic, the line through the two points of the curve fixed by $\rho(P)$ ; in the Coxeter graph, the unique vertex fixed by $\mu_C(P)$ . Composing these maps and their inverses gives the seam between any two of the five incarnations, and all of these seams commute.

Proof

If $P$ fixes $y$ , then $P\le G_y$ , which has order 6 and so is $N_G(Q)$ for its unique subgroup $Q$ of order 3; hence $Q=P$ and $G_y=N_G(P)$ . Since $N_G(P)$ is self-normalizing, exactly one point has this stabilizer. The map $P\mapsto y_P$ is a $G$ -map, since $gy_P$ is fixed by $gPg^{-1}$ , and a $G$ -map between transitive $G$ -sets of the same finite size is a bijection.

Example

The pair $\{0,\infty\}$ is fixed pointwise by $z\mapsto2z$ , of order 3, so the seam to the Sylow subgroups sends it to $\langle z\mapsto2z\rangle=\langle h\rangle$ , with $h\colon z\mapsto4z$ . Klein’s $\rho(h)$ permutes the coordinates cyclically; its fixed points on the curve are $(1:\omega:\omega^2)$ and $(1:\omega^2:\omega)$ , $\omega=e^{2\pi i/3}$ , and the line through them is $x+y+z=0$ , a bitangent since

x^3y+y^3z+z^3x\big|_{z=-x-y}=-(x^2+xy+y^2)^2.