marking

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

How are the symmetry groups of two theories compared?

An injective homomorphism from the reference group into the group a theory supplies; it makes the theory’s set a set acted on by the reference group.

Plate 1.3Two perfect matchings of

\Proj^1(\F_7)

: the antiflags with the common point 1 (gold) and those with the common line 246 (blue). The map

z\mapsto3z

, outside the group, carries one onto the other.

Definition(Marking)

Let a group $\Gamma$ act on a set $Y$ . A marking of $(\Gamma,Y)$ by $G$ is an injective homomorphism $\mu\colon G\to\Gamma$ . The marked set $Y_\mu$ is the $G$ -set with underlying set $Y$ and action $g\cdot y=\mu(g)y$ .

Two theories rarely share a group on the nose: the automorphism group of the Klein quartic and the group of the Fano plane are different groups that happen to be isomorphic, and a seam between incarnations in the two exists only after both groups are marked. The word is borrowed from Teichmüller theory, where a marked surface carries an identification of its fundamental group with a fixed reference group. It is unrelated to Burnside’s marks.

Lemma(Change of marking)

Let $\mu\colon G\to\Gamma$ be a marking and $\alpha\in\Aut(G)$ . (a) The stabilizer of $y$ in $Y_{\mu\alpha}$ is $\alpha^{-1}$ of its stabilizer in $Y_\mu$ ; so if $Y_\mu$ is transitive, $\st(Y_{\mu\alpha})=\alpha^{-1}(\st(Y_\mu))$ . (b) If $\alpha$ is inner, $\alpha(g)=hgh^{-1}$ , then $y\mapsto\mu(h)y$ is a $G$ -isomorphism $Y_\mu\to Y_{\mu\alpha}$ .

So an inner change of marking does not change the isomorphism class of the marked set, and an outer one moves the stabilizer class by that automorphism, changing nothing when the class is fixed by $\Aut(G)$ . The seams themselves do depend on the marking, even through inner automorphisms.

Proof

(a) $\mu(\alpha(g))y=y$ if and only if $\alpha(g)$ lies in the stabilizer of $y$ in $Y_\mu$ . (b) $\mu(h)\mu(g)y=\mu(hgh^{-1})\mu(h)y$ .

Theorem(Seams without markings)

Let $\Gamma\le\operatorname{Sym}(Y)$ and $\Gamma'\le\operatorname{Sym}(Y')$ be transitive permutation groups and $H$ a point stabilizer of $\Gamma$ . A permutation isomorphism is a bijection $f\colon Y\to Y'$ with $f\Gamma f^{-1}=\Gamma'$ .

(a) A permutation isomorphism $f$ determines $\theta_f\colon\gamma\mapsto f\gamma f^{-1}$ , and for every marking $\mu$ of $\Gamma$ , $f$ is a seam $Y_\mu\to Y'_{\theta_f\mu}$ . Conversely, if $\mu\colon G\to\Gamma$ and $\mu'\colon G\to\Gamma'$ are isomorphisms, every seam $Y_\mu\to Y'_{\mu'}$ is a permutation isomorphism $f$ with $\theta_f=\mu'\mu^{-1}$ .

(b) If there is a permutation isomorphism $Y\to Y'$ , the set of all of them is a torsor for the normalizer $N_{\operatorname{Sym}(Y)}(\Gamma)$ , and there is an exact sequence

1\longrightarrow\Aut_\Gamma(Y)\longrightarrow N_{\operatorname{Sym}(Y)}(\Gamma)\longrightarrow\Aut(\Gamma)_{[H]}\longrightarrow1,

where $\Aut(\Gamma)_{[H]}$ consists of the automorphisms that preserve the class $[H]$ . Hence $|N_{\operatorname{Sym}(Y)}(\Gamma)|=|N_\Gamma(H):H|\cdot|\Aut(\Gamma)_{[H]}|$ .

Proof

(a) $f(\mu(g)y)=\theta_f(\mu(g))f(y)$ , and conversely a seam satisfies $f\mu(g)f^{-1}=\mu'(g)$ for all $g$ .

(b) If $f,f'$ are permutation isomorphisms then $f^{-1}f'$ normalizes $\Gamma$ , and $fn$ is one whenever $n$ does. The kernel of $n\mapsto(\gamma\mapsto n\gamma n^{-1})$ is the centralizer of $\Gamma$ , which is $\Aut_\Gamma(Y)$ , and the image preserves $[H]$ because $n\Gamma_yn^{-1}=\Gamma_{ny}$ . Conversely, if $\alpha$ preserves $[H]$ , then $Y$ with the action $\gamma\ast y=\alpha(\gamma)y$ has the same stabilizer class, so the stabilizer principle gives a $\Gamma$ -isomorphism $n\colon Y\to Y^\alpha$ , which normalizes $\Gamma$ and induces $\alpha$ .

Corollary(Seams are markings)

Let $X$ be a rigid object of $G$ whose stabilizer class is fixed by $\Aut(G)$ , and let $\Gamma\le\operatorname{Sym}(Y)$ , $\Gamma'\le\operatorname{Sym}(Y')$ be permutation groups isomorphic to $G$ whose marked sets are incarnations of $X$ . Then $f\mapsto\theta_f$ is a bijection from the permutation isomorphisms $Y\to Y'$ onto the isomorphisms $\Gamma\to\Gamma'$ , and there are $|\Aut(G)|$ of them.

In words: for such an object, once the two symmetry groups are identified the identification of the two sets is forced, and every identification of the groups occurs.

Example

Among the orbital graphs of the object of size 28 are two families of seven $K_4$ ’s: in the antiflags, those with a common point and those with a common line. On the projective line each $K_4$ is four disjoint pairs covering $\Proj^1(\F_7)$ . Which family corresponds to the points of the Fano plane depends on the marking: the map $z\mapsto3z$ of $\PGL(2,7)$ , which lies outside $G$ and induces an outer automorphism, exchanges the two families.

The same marking fixes the labels $a$ and $b$ of the classes $V_4$ , $A_4$ and $S_4$ , and changing it by an outer automorphism exchanges them. The class of the object of size 28 is fixed by $\Aut(G)$ , so the normalizer of $G$ in the symmetric group of its 28 points has order $|N_G(S_3):S_3|\cdot|\Aut(G)|=336$ ; in the pairs it is $\PGL(2,7)$ .

Example

Two theories asked to agree can single out a class of markings. On a Coxeter edge, read in the antiflag model, the point rule and the line rule trace directed 4-cycles that give mutually inverse elements of order 4, and the bracket rule on the projective line orients the edge’s harmonic pair of pairs by the square class of $[a,c][c,b][b,a]$ . The bracket rule agrees with the point rule when the marking differs from $\mu_A$ by an inner automorphism, and with the line rule when it differs by an outer one. So a marking carries the stabilizers of the bisection $\{0,1,2,5\}\,|\,\{3,4,6,\infty\}$ to the stabilizers of points of the Fano plane if and only if the bracket rule agrees with the point rule: the square classes of $\F_7$ , which $G$ preserves and $\PGL(2,7)$ does not, tell the points of the Fano plane from its lines. The modular curve does the same for the inner class through its holomorphic structure.

Built from: objectstabilizer class
Builds: incarnationtwisting elementseam over an automorphismrefutedlifedouble lifeseam theoryGalois category
Objects: the twenty-eightthe seven pointsthe seven lines
In the Esquisse: 3La table des sutures du groupe d’ordre 1685Courte marche à travers la théorie de Galois10La table en deux, en sept et à l’infini17L’écart de Galois
In the volume: XThe Finite Celestial Sphere