Dictionnaire des sutures

0Le fonds classiqueThe classical stock

What does each theory supply before anything is compared?

Groups acting on sets, stabilizers, characters, and the classical groups with their geometries. The objects of the book are classical, and so is the group theory it uses: orbits and stabilizers, normalizers, automorphisms of permutation groups.

Several classical tools are adopted as they are: orbital graphs, which carry structure across seams; permutation isomorphisms; Gassmann equivalence; Burnside’s marks; power maps; the Frobenius–Schur indicator; equivariant bundles over a finite $G$-set; Hurwitz groups; the Bruhat–Tits building; and the triangle presentations of Cartwright, Mantero, Steger and Zappa. What the book isolates is the matchings themselves.

1L’incarnationIncarnation

When do two theories name one object?

An object of a group $G$ is a transitive $G$-set. A theory supplies a set and a group acting on it, both defined without reference to $G$; a marking identifies $G$ with a subgroup of that group, and the marked set is an incarnation of an object when it is $G$-isomorphic to it.

The floor rests on the stabilizer principle: an object is determined by its stabilizer class, so an entry of an atlas of objects is a conjugacy class of subgroups. The group of order 168 has exactly fifteen objects, and the Fano plane, the projective line over $\F_7$ and the Klein quartic each carry an incarnation of every one of them.

Its double cover $\SL(2,7)$ adds objects on which $-I$ acts without fixed points, sets that come from no set of the group of order 168. There are exactly four of these new objects, one over each class of subgroups of odd order.

• 1objectA transitive set of a group; up to isomorphism, a conjugacy class of its subgroups.
• 1stabilizer classThe conjugacy class formed by the stabilizers of an object’s points; it determines the object up to isomorphism.
• 1markingAn 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.
• 1incarnationA set acted on by the group, usually a theory’s marked set, that admits an equivariant bijection from the object.
• 1alignmentAn isomorphism from the object onto one of its incarnations; there are as many as the object has automorphisms.
• 11new objectA transitive set of the double cover $\SL(2,7)$ on which $-I$ acts without fixed points, so that it comes from no set of the group of order 168; there are exactly four, one over each class of subgroups of odd order.

2Les suturesSeams

In how many ways are two incarnations one, and do the ways agree?

A seam is a $G$-isomorphism between two incarnations of one object. The seams between two incarnations form a torsor under the automorphism group $N_G(H)/H$ of the object, so they are unique, and consistent around every cycle, exactly when the stabilizer is self-normalizing. Six of the fifteen objects of the group of order 168 are rigid in this sense.

The other nine carry freedom. When theories supply their seams by their own constructions, a cycle of natural seams can return a nontrivial automorphism, its monodromy. On the object of size 24 the flex-tangent map of the Klein quartic closes a cycle of length two with monodromy of order 3, and the power of an automorphism makes such monodromies comparable across theories.

Read in a choice of alignments, a family of seams over a graph is a lattice gauge connection with gauge group $N_G(H)/H$, and monodromy is its holonomy. A seam over an automorphism of $G$, a seam after twisting the action by it, joins incarnations that a fixed marking keeps apart, as the polarity joins the lines of the Fano plane to its points.

• 1seamAn equivariant bijection between two incarnations of one object; the seams between two incarnations form a torsor under the object’s automorphisms.
• 1seam groupoidThe groupoid whose vertices are a family of incarnations and whose arrows are their seams; consistency means it is the pair groupoid.
• 1rigid objectAn object with no automorphism but the identity; equivalently its stabilizers are self-normalizing, and then every seam is unique.
• 1coherenceA family of seams is coherent when every route between two incarnations gives the same map; automatic for rigid objects, and otherwise the same as coming from one choice of alignments.
• 4seam systemA family of incarnations with a chosen set of seams among them, loops allowed: typically the natural identifications that the theories provide.
• 4seam monodromyThe composite of seams around a closed walk, an automorphism of the incarnation; it measures how far a family of seams is from one choice of alignments.
• 5Galois categoryA category of finite sets with an action, and a functor that forgets the action: the objects of a group are its connected objects, seams its isomorphisms, markings identifications of fibre functors, and seams over automorphisms its twists by Out(G).
• 4gaugeA choice of alignments, one for each incarnation of a seam system over a graph; it turns the seams into link variables in $N_G(H)/H$, so that a seam system is a lattice gauge connection and its monodromy is holonomy.
• 4powerFor a self-centralizing cyclic stabilizer, the residue k such that every seam to a conjugacy class turns the automorphism into the k-th power map; it depends on no seam, class or marking.
• 4quotient classThe stabilizer class of the quotient of an incarnation by an automorphism; seams preserve it, and it names the three involutions of the object of size 84.
• 10twisting elementThe element c by which a symmetry conjugates the marking; correcting the symmetry by c gives an automorphism, its equivariant twist, whose power is inverse to that of c.
• 1seam over an automorphismA bijection that carries the action of each element to the action of its image under an automorphism of the group; a seam is a seam over the identity, and a bridge refuted for one marking can be built over an outer automorphism.

3Ce qui est suWhat is known

What has been proved about a bridge, and what is proved not to exist?

A bridge asserts that two sets, given in two theories, are incarnations of one object, and its status records what is known: built, type, name or refuted. Only built bridges are theorems. For two marked sets of one group the stabilizer principle decides every type bridge, which is either built or refuted. Type recurrence is not identification.

Beside the statuses stands the negative space: absences, theorems that something is not there, with their windows, the parameter values where the excluded thing can still happen, and their imprints, the structures an absence forces to exist. The absences reduce to one another in three clusters: the group of order 168, the octonions and Hilbert space.

A map between theories that is not a seam is a description: it goes one way and forgets something, and what it forgets at a point is its kernel, a stabilizer. Read so, each concept of the floor is a statement about a description and what it forgets: 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.

• 1bridgeThe assertion that two sets, given in two theories, are incarnations of one object; only a built bridge is a theorem.
• 1statusBuilt, type, name or refuted: a record of what is known about a bridge, not a property of the objects.
• 2refutedThe status of a bridge proved false: no seam exists for the markings in question. A type recurrence proved to be no identification.
• 1descriptionA surjective equivariant map between sets on which one group acts: it goes one way and may forget something, and it forgets nothing exactly when it is a seam.
• 1kernelThe stabilizer of the image of a point under a description: what the description cannot tell apart there. It covers the kernel of a homomorphism, the stabilizer of an orbit and the congruence kernel of a reduction.
• 2absenceA theorem that a collection of structures, specified by explicit axioms, has no member with a stated property; it marks where the atlas cannot be stitched.
• 3forced gapA class of subgroups that no basic figure of a theory has as its stabilizer class; in the seam table every forced gap is filled by a composite figure, and only the Coxeter graph reaches every class with its simplest figures.
• 17Galois gapThe number of conjugacy classes minus the number of rational classes: the dimension of the class functions that no combination of finite G-sets reaches, measured by the same Galois orbits that decide which twists no count can hear.
• 2windowFor a graded absence, the set of parameter values that actually occur; it is usually small, and its edges carry the structure.
• 2imprintA structure that exists, with a theorem characterizing it by an absence: terminal (the survivors at the edge of a window), carrier (what carries local data that do not globalize) or separating (a finer invariant).
• 2reductionAn absence reduces to another when the book proves it from the other without reproving it; the reductions sort the absences into three clusters that meet only through bridges.

4Les doubles viesDouble lives

When does one group carry the geometries of two families?

A life of a group is an isomorphism onto a member of the families $\PSL(n,q)$, acting on its projective space, or $A_m$, acting on $m$ letters. Isomorphisms between members of different families are rare: by Artin’s absence exactly four groups have a double life, $A_5$, $\PSL(2,7)$, $A_6$ and $A_8$. The floor consists of those survivors, read as seams.

Each double life comes with a dictionary of which natural sets of the two lives are one object, computed by matching stabilizers. For the group of order 168 the dictionary is the seam table itself. In three of the four double lives an outer automorphism exchanges two dual objects of one life and is unremarkable in the other.

• 6lifeAn isomorphism of a group onto a member of the families PSL(n,q) or A_m: a marking whose target brings a classical geometry with it.
• 6double lifeTwo lives of one group in different members of the families; by Artin’s absence exactly four groups have one: $A_5$, $\PSL(2,7)$, $A_6$ and $A_8$.
• 6type lawA twisted Galois symmetry of a lattice is seen on its residues by how the prime decomposes: as a seam between two residues where the prime splits, semilinearly where it is inert, linearly where it ramifies.
• 6dictionaryFor each object of a group with a double life, the natural sets of each life that are incarnations of it, computed by matching stabilizers.

5Les complétionsCompletions

Where do the finite geometries sit inside buildings over local fields?

Each life of a double life is a geometry over a finite field $\F_p$, the residue field of $\Q_p$. The geometry is the link of a vertex of the Bruhat–Tits building over $\Q_p$, and the stabilizer of the vertex acts on it through the finite group. A completion of a finite projective geometry is such a building with such a vertex.

So a double life sits at two vertices: the group of order 168 acts on the Heawood graph, the link of a vertex of the building of $\PGL(3,\Q_2)$, and on $\Proj^1(\F_7)$, the link of a vertex of the tree of $\PGL(2,\Q_7)$. The octonion multiplication table glues Fano links into the building of $\PGL(3)$ over $\F_2(\!(t)\!)$, and no subgroup of finite index of its group is isomorphic to one of Mumford’s lattice.

Kato’s hermitian form glues the same links, without symmetry, into the building over $\Q_2$, as Mumford’s lattice; a gluing that a Frobenius group of order 21 respects is the octonion one, and it lives in characteristic 2. At 7 Mumford’s form has its own tree, whose base link is the sky, and over $\Z[1/14]$ the group of order 168 is the stabilizer of a vertex, Klein’s lattice, at which it carries both lives.

At Klein’s lattice the finite geometry is found among short vectors and neighbours: the stabilizer of a point acts on the neighbour through it as the rotations of a cube, a flag is a pair of vectors of norm 2 whose reflection is a half-turn of that cube, and an antiflag is one of its diagonals, of norm 3. One step beyond the link the two trees at 7 carry a doublet and its symmetric square; the object of the points of the Fano plane is carried along the one tree in exactly one way and along the other in none; and the two parents carry independent flips, the sign changes of $\sqrt{-3}$ and $\sqrt{-7}$, of which only the first is seen by the oriented cells of the link complement.

The two parents are joined only by fiber products: across scales none keeps the finite line attached, and at one scale the attachment is forced. Around the loops of the scale tree a single relabelling carries the signed octonion table without reversals exactly on the Iwahori subgroup, and on seven loops in eight it must reverse two units; carried observer by observer, every loop returns each fiber changed only by colour, consistently with the meetings.

• 9completionA building over a local field with a vertex whose link is the flag complex of a finite projective geometry; the finite geometry lives over the residue field.
• 13orientationEach arithmetic parent of the group of order 168 carries an orientation: the sign change of $\sqrt{-3}$ turns the congruence link complement into its mirror image, and that of $\sqrt{-7}$ exchanges the octonion table with its Weil mirror. The two flips are independent, and once the signs of the units at the cusps are treated as a convention, only the first is seen by the structures of the link complement.

6Les continusContinua

How do the finite objects reappear in real and complex geometry?

A finite object can appear in a continuous geometry in two ways: as a configuration of points fixed in place by a finite group of symmetries, an embedded continuum, or as a set of classes of an arithmetic configuration modulo a congruence subgroup, an arithmetic one. Which kind an object can have is decided by absences.

The archimedean place joins this floor to the completions: the congruence that gives a residue field at a prime gives, over $\C$, Thurston’s congruence link complement, whose eight cusps are the points of $\Proj^1(\F_7)$ and whose cells are objects of the group of order 168. The projective line $\Proj^1(\F_7)$ has no embedded continuum in $\Proj^1(\C)$ or in Klein’s plane, only this arithmetic one.

Every row of the seam table is a configuration of cells of that manifold, and the Fano incidence among them is the absence of a shared face. In the Cayley plane one point and one imaginary unit carry the intersection of two maximal subgroups of $F_4$ found by Todorov and Dubois-Violette; in its complexification the same point carries the $\mathbf{16}$ of $\mathfrak{so}(10)$.

On the link complement the spinor system, the local system of the defining representation of $\SL(2,\Z[\omega])$, carries the first of the two parents’ flips at the cusps, and the operators that move between pairs of cusps generate a Clifford algebra whose centre is a single sign.

• 14continuumA homogeneous space of a Lie group that carries the object, either as an equivariant configuration (embedded) or as classes modulo a congruence subgroup (arithmetic).
• 13spinor systemThe local system $V$ on the congruence link complement given by the defining representation of $\SL(2,\Z[\omega])$. Its boundary scattering is one constant times the Paley matrix; its cusp lines transform as $V$ and not as its mirror $\bar V$; and the moves between pairs of cusps keep that class.
• 13commit algebraAt a unit $e_p$ of the octonions, the algebra generated by the six left multiplications by the other units, each tensored with a flip of a two-state counter, and by the counter’s sign. It is the complex Clifford algebra $\mathrm{Cl}_7$, a sum of two matrix algebras, and its centre is spanned by the identity and one sign, the chirality $D=-iL_{e_p}$ read with the parity of the number of moves.

Le sujetThe subject

What is studied?

Seam theory, the study of seams: whether incarnations in different theories can be joined, in how many ways, whether the joins are consistent around a cycle of theories, how they depend on the identification of symmetry groups, what the maps that are not joins forget, and where a join is impossible. The objects are classical; the joins, not the pieces, are at the centre.

The last question gives the subject its shape: a table of objects against theories in which some cells are empty by necessity, a negative space bounded by the seams that do exist.

• 1seam theoryThe study of seams: when they exist, how many there are, whether they are consistent, how they depend on markings, what the maps that are not seams forget, and where seams are impossible.