Un objet, plusieurs noms

Objets, incarnations, suturesObjects, incarnations, seams

Throughout, a group $G$ acts on the left. A $G$-set is a set $X$ with an action $(g,x)\mapsto gx$; a $G$-map $f$ satisfies $f(gx)=gf(x)$; $\Iso_G(X,Y)$ and $\Aut_G(X)$ are the $G$-isomorphisms $X\to Y$ and $X\to X$. The stabilizer of $x$ is $G_x$; the cosets $G/H$ form a transitive $G$-set in which the coset $H$ has stabilizer $H$; and $N_G(H)$ is the normalizer of $H$.

An object of $G$ is a transitive $G$-set: the word records what several theories name. 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. An incarnation of an object $X$ is a $G$-set $Y$, usually a theory’s set with its group marked by $G$, for which some $G$-isomorphism $X\to Y$ exists; such an isomorphism is an alignment, which lays the incarnation over the object point by point.

A seam between two incarnations of one object is a $G$-isomorphism between them. An incarnation must have an alignment, but none 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 the chapter answers, whether a seam is forced, would be assumed away. The answer is that alignments, and with them seams, carry exactly as much freedom as the object has automorphisms.

Le principe du stabilisateurThe stabilizer principle

Stabilizers move by conjugation, so the stabilizers of an object’s points form one conjugacy class of subgroups, its stabilizer class $\st(X)$. The stabilizer principle says that this single datum decides everything: two objects are isomorphic exactly when their classes agree; an alignment is fixed by choosing where one point goes, among the points with the same stabilizer; and the automorphisms of $G/H$ form the group $N_G(H)/H$.

On the projective line, the stabilizer of the pair $\{0,\infty\}$ consists of the six maps $z\mapsto\lambda z$ and $z\mapsto-\lambda/z$ with $\lambda\in\{1,2,4\}$, a group $S_3$. It is its own normalizer, as every subgroup of order six of the group of order 168 is, so the object of the 28 pairs has only the identity as automorphism, and between it and any other incarnation of it there is exactly one seam.

The principle fits in one sentence. The groupoid whose vertices are the objects of $G$ and whose arrows are the $G$-isomorphisms is equivalent to the disjoint union, over the conjugacy classes $[H]$ of subgroups, of the groups $N_G(H)/H$, each a groupoid with one vertex. An entry of an atlas of objects is therefore a conjugacy class of subgroups, and the residual freedom in matching its incarnations is $N_G(H)/H$. Parts (b) and (c) are classical; in the language of permutation groups, $\Aut_G(X)$ is the centralizer of $G$ in $\operatorname{Sym}(X)$.

Objets rigides, sutures cohérentesRigid objects, coherent seams

An object is rigid if its only automorphism is the identity, rigid in the combinatorial sense. By the principle this happens exactly when its stabilizer is self-normalizing, and then every incarnation has one alignment and any two incarnations one seam. Uniqueness forces consistency: if $s_{ij}$ is the only seam $Y_i\to Y_j$, then $s_{jk}\circ s_{ij}$ and $s_{ik}$ are both seams $Y_i\to Y_k$, so they are equal. The seam groupoid of a rigid object is the pair groupoid, with exactly one arrow from each vertex to each vertex.

When the object is not rigid, every vertex group of the seam groupoid is $N_G(H)/H\neq1$, and there are $|N_G(H):H|$ seams between any two incarnations. Such seams are usually chosen one at a time, each by a construction natural in its own pair of theories, and around a cycle $Y_1\to Y_2\to\cdots\to Y_1$ the choices compose to an automorphism of $Y_1$. Through an alignment it is an element of $N_G(H)/H$, well defined up to conjugation, which measures how far the chosen seams are from consistent. This composite is the monodromy of the cycle, the subject of Chapter 4, where a family of seams over a graph turns out to be a lattice gauge connection with gauge group $N_G(H)/H$.

MarquagesMarkings

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. A marking of $(\Gamma,Y)$ by $G$ is an injective homomorphism $\mu\colon G\to\Gamma$, and the marked set $Y_\mu$ is $Y$ with $g\cdot y=\mu(g)y$. The word is borrowed from Teichmüller theory, where a marked surface carries an identification of its fundamental group with a fixed reference group; it has nothing to do with Burnside’s marks.

Changing a marking by $\alpha\in\Aut(G)$ replaces each stabilizer by its image under $\alpha^{-1}$. An inner change, $\alpha(g)=hgh^{-1}$, leaves the marked set’s isomorphism class alone, since $y\mapsto\mu(h)y$ is a $G$-isomorphism $Y_\mu\to Y_{\mu\alpha}$; an outer change moves the stabilizer class by $\alpha$, and changes nothing when the class is fixed by $\Aut(G)$. The seams themselves depend on the marking, even through inner automorphisms. The marking-free notion is classical: a permutation isomorphism from $\Gamma\le\operatorname{Sym}(Y)$ to $\Gamma'\le\operatorname{Sym}(Y')$ is a bijection $f$ with $f\Gamma f^{-1}=\Gamma'$, and when the markings are isomorphisms the seams $Y_\mu\to Y'_{\mu'}$ are exactly the permutation isomorphisms with $f\mu(g)f^{-1}=\mu'(g)$. The permutation isomorphisms $Y\to Y'$ form a torsor for $N_{\operatorname{Sym}(Y)}(\Gamma)$, of order $|N_\Gamma(H):H|\cdot|\Aut(\Gamma)_{[H]}|$.

In the twenty-eight, the antiflags with a common point and those with a common line form two families of seven complete graphs $K_4$; on the projective line each $K_4$ is a partition of the eight points into four pairs. Which family belongs to the points of the Fano plane depends on the marking: $z\mapsto3z$, which lies in $\PGL(2,7)$ but not in $G$ and induces an outer automorphism, exchanges the two families.

Ponts et statutsBridges and their status

A bridge is the assertion that two sets, given in two theories, are incarnations of one object, and its status records what is known about it. It is built when the symmetry groups are marked by $G$ and a seam has been written down and proved to be a $G$-map; of type when the two sides are known to share some invariant of objects (the group, the number of elements, a stabilizer up to abstract isomorphism, the permutation character) but no seam has been written down; a name when they share a name and nothing more is known. A status describes knowledge, not the objects: a bridge of status type or name may later be built, or shown to be false.

For two marked sets of one group the stabilizer principle decides every type bridge. Either the stabilizer classes agree, and $gx\mapsto gy$ is a seam for any pair of points with $G_x=G_y$, so the bridge is built as soon as one such pair is exhibited; or they differ, no seam exists, and the bridge is refuted. A refuted bridge marks a cell of the atlas that is empty by necessity, the subject of Chapter 2.

The strongest invariant short of the stabilizer class that a type bridge usually records is the permutation character, and it does not suffice. Subgroups with one permutation character are called Gassmann equivalent. Perlis used the point and line stabilizers of $\GL(3,2)$ to construct non-isomorphic number fields of degree 7 with the same Dedekind zeta function, and through Sunada’s method Gordon, Webb and Wolpert used them to construct plane domains that are isospectral but not congruent. Type recurrence is not identification. Chapter 17 says exactly which twists counting cannot hear, those that act on the classes of $G$ as Galois acts, and the polarity below is one of them.

Sutures au-dessus d’un automorphismeSeams over an automorphism

The polarity is not a seam, but it becomes one once the action on its target is twisted. The twist $Y^\alpha$ of a $G$-set by $\alpha\in\Aut(G)$ is the same set with the action $g\ast y=\alpha(g)y$, and a seam over $\alpha$ from $X$ to $Y$ is a bijection $f$ with $f(gx)=\alpha(g)f(x)$; a seam is a seam over the identity. The polarity is a seam over the outer automorphism $g\mapsto(g^{\mathsf T})^{-1}$ from the lines to the points: refuted over the identity, built over $\alpha$. This is the precise sense of the phrase, used throughout the later chapters, that a refuted bridge becomes a seam after twisting by the outer automorphism. Over an inner automorphism nothing new happens: a symmetry $\sigma$ with $\sigma\mu(x)\sigma^{-1}=\mu(cxc^{-1})$ is a seam over $x\mapsto cxc^{-1}$, and $\mu(c)^{-1}\sigma$ is a seam.

An example comes from the double cover. In the Weil representation of $\SL(2,7)$, Chapter 11 builds two families of algebras $\mathfrak{su}(3)$, one algebra of each family over each of the 28 pairs $o$ of points of $\Proj^1(\F_7)$, the second belonging to the mirror of the octonion table. Each family is an incarnation of the object of size 28, which is rigid, with stabilizer class $S_3$ fixed by $\Aut(G)\cong\PGL(2,7)$. So the seam $\Phi\colon\mathfrak f_o\mapsto\mathfrak f'_o$ over the identity is unique; conjugation by an element $m$ of $\PGL(2,7)$ outside $G$ carries $\mathfrak f_o$ onto $\mathfrak f'_{m(o)}$, a seam over the outer automorphism $g\mapsto mgm^{-1}$ and not a seam; and over each pair, $\Phi$ agrees with conjugation by the polarity that fixes the pair. The seam over the identity is assembled from 28 seams over outer automorphisms, each correct at its own pair.

Les vingt-huitThe twenty-eight

From here on $G=\PSL(2,7)$, the 168 Möbius maps $z\mapsto(az+b)/(cz+d)$ of $\Proj^1(\F_7)=\F_7\cup\{\infty\}$ with $ad-bc=1$, generated by $g\colon z\mapsto z+1$, $h\colon z\mapsto4z$ and $s\colon z\mapsto-1/z$. It is classical that $G$ is simple, that $G\cong\GL(3,2)$, and that $\operatorname{Out}(G)$ has order 2, so $|\Aut(G)|=336$; all three were also checked by machine. Working in $\GL(3,2)$, the 56 elements of order 3 form one class with centralizers of order 3, so there is no element of order 6; the 28 subgroups of order 3 are the Sylow 3-subgroups, each with normalizer $S_3$; and every subgroup of order 6 is the normalizer of its unique subgroup of order 3, so the subgroups of order six form one self-normalizing class of 28.

Five incarnations. (P) The two-element subsets of $\Proj^1(\F_7)$, where the stabilizer of $\{0,\infty\}$ is $\{z\mapsto\lambda z,\ z\mapsto-\lambda/z\}$. (S) The Sylow 3-subgroups under conjugation, with no marking needed. (A) The antiflags $(p,L)$ of the Fano plane, with stabilizer $\GL(p)\times\GL(L)\cong S_3$, since $V=p\oplus L$. (B) The bitangents of the Klein quartic $x^3y+y^3z+z^3x=0$, marked by Klein’s representation $\rho$: the line $x+y+z=0$ is one, because the quartic restricted to it is $-(x^2+xy+y^2)^2$, and its stabilizer is $\langle h,s\rangle$. (C) The vertices of the Coxeter graph, marked onto the derived subgroup of its automorphism group of order 336.

The seams are explicit. Each Sylow 3-subgroup $P$ fixes exactly one point of any incarnation, the point with stabilizer $N_G(P)$: on the line, the pair of points $P$ fixes; in the Fano plane, the antiflag of the unique point and the unique line it fixes; on the quartic, the line through the two points of the curve it fixes, which for $P=\langle h\rangle$ are $(1:\omega:\omega^2)$ and $(1:\omega^2:\omega)$, spanning $x+y+z=0$; in the graph, its one fixed vertex. Composing these maps gives the seam between any two of the five, and all of them commute.

Le graphe de Coxeter est intrinsèqueThe Coxeter graph is intrinsic

The seams carry structure as well as points. The point stabilizers of the twenty-eight have orbits of lengths 1,3,3,3,6,6,6, so $G$ has three cubic orbital graphs on it. Two are disjoint unions of seven complete graphs $K_4$; the third is connected, distance-regular with intersection array $\{3,2,2,1;1,1,1,2\}$, and so is the Coxeter graph. A seam is a $G$-isomorphism, so it carries orbital graphs to orbital graphs, and every incarnation carries the Coxeter graph.

Read in the models: on the line, two pairs are adjacent when they are disjoint and their cross-ratio is $-1$, and since $(0,\infty;c,d)=c/d$ the neighbours of $\{0,\infty\}$ are $\{1,6\}$, $\{2,5\}$ and $\{3,4\}$. In the Fano plane, $(p,L)$ and $(q,M)$ are adjacent when $p\neq q$, $L\neq M$, $p\notin M$ and $q\notin L$, equivalently when their triangles $\mathcal P\setminus(L\cup\{p\})$ are disjoint. In the group, $P$ and $Q$ are adjacent when $tu$ has order 4 for all elements $t\in P$ and $u\in Q$ of order 3.

The two families of $K_4$‘s are the two ways the twenty-eight sees the objects of size 7: the antiflags with a common point and those with a common line. By the theorem on seams without markings, the normalizer of $G$ in the symmetric group of the 28 points has order $|N_G(S_3):S_3|\cdot|\Aut(G)|=336$. On the pairs it is $\PGL(2,7)$: it permutes the orbital graphs, so it preserves the one connected cubic graph, which has exactly 336 automorphisms.

Descriptions et noyauxDescriptions and their kernels

A seam identifies two incarnations and forgets nothing. Most maps between theories are not seams: they go one way and forget something. Reduction modulo a prime, the passage from a double cover to its quotient, and the passage from a group to one of its orbits all lose information, and in each case the loss is a subgroup. A description is a surjective $\Gamma$-map $d\colon X\to Y$; its kernel at $x_0$ is $K_d(x_0)=\{\gamma\in\Gamma: d(\gamma x_0)=d(x_0)\}=\Gamma_{d(x_0)}$, and what it forgets there is the fibre $d^{-1}(d(x_0))$, the orbit $K_d(x_0)x_0$ when $X$ is transitive. A description forgets nothing exactly when it is a bijection: a description that forgets nothing is a seam.

For a surjective homomorphism, read as a description of the group by its image, the kernel at 1 is the kernel; for the orbit map $\gamma\mapsto\gamma y$ it is $\Gamma_y$, so the stabilizer principle says that an object is the regular set with the kernel of its orbit description divided out; between transitive sets with stabilizers $H\le L$ the kernel is $L$, and the description forgets a copy of $L/H$. None of this is new. A central kernel of prime order acts on every transitive set trivially or without fixed points, and for $\SL(2,7)\to\PSL(2,7)$ the objects the kernel $\{\pm I\}$ cannot see are the four new objects of Chapter 11, on which it acts without fixed points.

Two more kernels. The derivations of the octonions form $\mathfrak g_2$, of dimension 14; describing a derivation by its effect on one imaginary unit $e_x$ maps $\mathfrak g_2$ onto the 6-dimensional complement of $\operatorname{span}(1,e_x)$, and the kernel is an algebra $\mathfrak{su}(3)_x$ of dimension 8 commuting with left multiplication by $e_x$. And a structure carried by an orbit is induced from a kernel: by Frobenius reciprocity, $\operatorname{Ind}_H^GW$, the sections of the bundle $G\times_HW$ over $G/H$, is the universal structure over the orbit carrying data $W$ given on the kernel. The permutation representation, the spin bundle of Chapter 2 and the family of algebras $\mathfrak{su}(3)$ over the 28 pairs are all of this kind.

Le troisième étage par les noyauxFloor three through kernels

The concepts that record what is known about seams (statuses, absences, imprints and monodromy) can all be stated as facts about descriptions. Each statement is a reformulation of classical facts, and no novelty is claimed. Statuses: a description between transitive sets exists exactly when a stabilizer of the source lies in a stabilizer of the target, and it forgets nothing exactly when it is a seam. So a bridge is built exactly when there is a description that forgets nothing, and between sets of one size it is refuted exactly when there is no description at all.

Absences: send each point of a finite $G$-set to the class of its stabilizer, its orbit type. The fibre over $[H]$ has $|G:H|\,c_H(Z)$ points, where $c_H(Z)$ counts the orbits isomorphic to $G/H$, and these numbers are recovered from the marks by inverting the table of marks of Chapter 3. So a forced gap is an empty fibre of the orbit-type description, detected by the marks, and the window of an absence is the image of its invariant. Monodromy: read the holonomy of a seam system over a graph as a description of the fundamental group; its kernel $N$ is the group of loops around which the seams close up, the system is coherent exactly when $N$ is everything, and on the covering with fundamental group $N$ the pulled-back system becomes coherent, every other such covering covering this one.

In one line each: a built bridge is a description that forgets nothing; an absence is an empty fibre; a carrier imprint is induced from what an orbit description forgets; and monodromy is what remains of the loops once the kernel of the holonomy is divided out.