alignment

Floor 1, L’incarnation · introduced in Chapter 1, Un objet, plusieurs noms

How is an incarnation laid over its object, point by point?

An isomorphism from the object onto one of its incarnations; there are as many as the object has automorphisms.

Plate 1.5The 24 nonzero vectors of

\F_7^2

up to sign, three on each of eight spokes. The three alignments of the object of size 24 differ by scaling, which turns the three vectors of every spoke around together.

Definition(Incarnation, alignment)

An alignment of an incarnation $Y$ of an object $X$ is a $G$ -isomorphism $X\to Y$ . An alignment lays the incarnation over the object, point by point; the word is chosen for that picture.

An incarnation is required to have an alignment, but no alignment is fixed. If alignments $\varphi\colon X\to Y$ and $\varphi'\colon X\to Y'$ were part of the data, the two incarnations would come with the preferred seam $\varphi'\circ\varphi^{-1}$ , and the question whether a seam is forced would be assumed away. Alignments, and with them seams, carry exactly as much freedom as the object has automorphisms.

Theorem(The stabilizer principle)

Let $x\in X$ and $H=G_x$ . Evaluation at $x$ is a bijection from the alignments $X\to Y$ onto $Y_H=\{y\in Y: G_y=H\}$ , the alignment with value $y$ being $gx\mapsto gy$ . The alignments form a torsor for $\Aut_G(X)\cong N_G(H)/H$ , so an incarnation has exactly $|N_G(H):H|$ of them.

Proof

A $G$ -map from $X=Gx$ is determined by its value at $x$ , and an isomorphism preserves stabilizers. If $G_y=H$ , then $gx\mapsto gy$ is well defined, equivariant and bijective. Two alignments $f,f'$ differ by the automorphism $f^{-1}f'$ of $X$ .

Example

The object of size 24 has automorphism group $N_G(C_7)/C_7\cong C_3$ . So each of its incarnations, such as the 24 flexes of the Klein quartic or the 24 nonzero vectors of $\F_7^2$ up to sign, has three alignments, and between any two incarnations there are three seams. Scaling the vectors by $\lambda\in\F_7^\times$ is one of these automorphisms.

Built from: incarnationobject
Builds: seamrigid objectcoherenceseam monodromygauge
Objects: the object of size 24the twenty-eight
In the volume: XThe Finite Celestial Sphere