incarnation

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

When is a set in some theory a form of a given object?

A set acted on by the group, usually a theory’s marked set, that admits an equivariant bijection from the object.

{0, ∞}(1, 246)d₀⟨z ↦ 2z⟩

Projective line

a 2-subset of P¹(F₇)

Fano plane

an antiflag (p, L), p ∉ L

Graphs

a vertex of the Coxeter graph

The group

a Sylow 3-subgroup of PSL(2,7)

generator: z ↦ 2z
on the eight points: (1 2 4)(3 6 5)
fixes: {0, ∞}, and nothing else

Klein quartic

a bitangent of x³y + y³z + z³x = 0

the line: x + y + z = 0
touching: at (1 : ω : ω²) and (1 : ω² : ω), the two points fixed by ρ of the subgroup

Choose a vertex of the Coxeter graph, or step through all twenty-eight.

Plate 1.4One element of the object of size 28 in five theories: the pair

\{0,\infty\}

, the Sylow subgroup

\langle z\mapsto2z\rangle

, an antiflag, the bitangent

x+y+z=0

and a vertex of the Coxeter graph.

Definition(Incarnation, alignment)

Let $X$ be an object of $G$ . An incarnation of $X$ is a $G$ -set $Y$ , usually a marked set supplied by some theory, for which there exists a $G$ -isomorphism $X\to Y$ . Such an isomorphism is an alignment of the incarnation.

A theory, for this purpose, is a body of mathematics (projective geometry over $\F_2$ , the geometry of a plane curve, the subgroup structure of a group, graph theory) that supplies a set $Y$ and a group $\Gamma$ acting on it, both defined without reference to $G$ . The notion is not formalized, and nothing proved depends on where $Y$ and $\Gamma$ come from.

Theorem(The first block) computed

For the group of order 168, natural sets of figures in five theories (the Fano plane, the projective line over $\F_7$ , the group itself, the Klein quartic, and the Coxeter and Heawood graphs) were split into orbits. Every orbit listed in the seam table is a transitive $G$ -set whose stabilizers form the class of its row. In particular the Fano plane, the projective line and the Klein quartic each carry an incarnation of every one of the fifteen objects.

Any two orbits with the same class are joined by the explicit seam $gx\mapsto gy$ , for points with equal stabilizers, so every entry of the table has status built.

Proof

By machine, in exact arithmetic: each set of figures was built, split into orbits under $G$ acting through the marking of its theory, and the stabilizer of a representative of each orbit was identified among the 179 subgroups. Many entries also have short proofs; for instance an ordered pair of points $(p,q)$ of the Fano plane determines the line $L=pq$ , and its stabilizer is the group of elations with axis $L$ , a member of $V_4^b$ .

Example

The object of size 28 has an incarnation in each theory: the 28 two-element subsets of $\Proj^1(\F_7)$ , with the natural action; the 28 Sylow 3-subgroups of $G$ under conjugation, where no marking is needed; the 28 antiflags of $\mathrm{PG}(2,2)$ , marked by an isomorphism $\mu_A\colon G\to\GL(3,2)$ ; the 28 bitangents of the Klein quartic $x^3y+y^3z+z^3x=0$ , marked by Klein’s representation $\rho\colon G\to\SL(3,\C)$ ; and the 28 vertices of the Coxeter graph, marked by an isomorphism onto the derived subgroup of its automorphism group.

Once the symmetry groups are identified, an element has exactly one name in each of the five theories, and the names agree along every route between them.

Remark(The table closed)

The closed seam table has seven columns: the Fano plane, the projective line, the group, the Klein quartic, the Coxeter and Heawood graphs, the octonions, and the cells of Thurston’s congruence link complement $M$ . Every entry is built: each column carries an incarnation of each of the fifteen objects, and the entries of a row are joined by explicit seams. What distinguishes the columns is which classes their simplest figures can reach, the forced gaps. Over the four rows of odd order the double cover $\SL(2,7)$ adds a new object, and three of the four have a second theory in its Weil representation.