00Generative identity · third edition · A90921DD

Seam
Theory

51 marks, one for each of the 15 objects and the 36 concepts, each computed from its own mathematics. Nothing is seeded, so nothing reseeds: a mark changes only when its mathematics does.

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.

seam theory · roof · arcs 20 · core S3 C7 7:3 S4a S4b

01Principles

Six questions, six rules

Seams states the subject as six questions. Each row gives the book’s answer, the arc of the roof’s mark that answers it, and the rule by which every object’s mark shows it.

Q1
Existence
stabilizer class
Two sets are joined exactly when their stabilizer classes agree.
1.2 stabilizer class
Teeth
A mark depends on the stabilizer class and nothing else. Its outer ring has a tooth for each point of $G/H$ , each $|H|$ steps of the field of 168, so the teeth always close the circle. Every incarnation of an object wears that object’s one mark.
$G/C_7$ · 24 teeth × 7 steps = 168
Q2
Number
$N_G(H)/H$
The seams between two incarnations form a torsor under $N_G(H)/H$ .
2.1 seam
Rotor
One blade for each automorphism, $|N_G(H):H|$ of them: as many as the seams between any two incarnations. The fifteen objects need six rotors.
- 1
- 2
- 3
- 4
- 6
- 168
Q3
Consistency
rigidity, monodromy
Seams are unique and coherent exactly when the object is rigid, and otherwise natural seams can carry monodromy.
2.3 rigid object · 2.6 seam monodromy
Gold
The identity’s blade is gold. A rigid object has no other, and its hub is filled gold: between any two of its incarnations the seam is forced. Monodromy turns the gold blade (05).
- $S_3$
- $D_8$
- $7{:}3$
- $S_4^a$
- $S_4^b$
- $G$
Q4
Dependence on markings
$\operatorname{Out}(G)$
An inner change of marking changes nothing up to isomorphism, while an outer automorphism moves the stabilizer class, as it exchanges the points and the lines of the Fano plane, and a bridge refuted for one marking is built over the outer automorphism.
1.3 marking
Mirror
The marking shears the teeth of an $a$ -class to the left and of a $b$ -class to the right, and flies its pennant to that side. A mirror turns each $a$ -mark into its $b$ -mark, as the outer automorphism exchanges the classes; the other nine marks are their own mirror images.
- fixed
- class a
- class b
- a and b
Q5
Forgetting
description, kernel
A map between theories that is not a seam is a description, and what it forgets at a point is its kernel, a stabilizer.
3.5 kernel
Rings
Each inner ring is a description: the map $G/H\to G/K$ of a cover, which goes one way and forgets. Its kernel at the point $H$ is $K$ , the stabilizer of the image, and what it forgets there is a copy of $K/H$ : the $|K:H|$ teeth that one tooth of the ring gathers.
$G/C_7\to G/(7{:}3)$ · one tooth forgets 3
Q6
Impossibility
negative space
The negative space of absences, with their windows and imprints.
3.6 absence
Pairs
Exactly three pairs of objects share a permutation character, and every bridge between the two of a pair, for one marking, is refuted. Their marks agree in every count the grammar draws and differ only by the mirror; over the outer automorphism, the refuted bridge is built.
- $A_4^a$ $A_4^b$
- $S_4^a$ $S_4^b$
- $V_4^a$ $V_4^b$

02Construction

One mark, constructed

The object of size 24, whose stabilizer is $C_7$ , drawn on its grid. Every mark of the edition is built on the same square of 25 units, the same rings and the same field; only the inputs change.

Sheet 02.1 · construction1 : 1 · 25 units

1
Teeth. $|G:H|=24$ , each $|H|=7$ steps of the field: $24\times7=168$ .
2
Map. One ring for the cover $G/C_7\to G/(7{:}3)$ , a description: 8 teeth, each gathering $|7{:}3:C_7|=3$ points.
3
Rotor. $|N_G(C_7):C_7|=3$ blades; the hub is open, as the object is not rigid.
4
Identity. The gold blade at twelve o’clock.
5
Staff. Every mark’s index, on the axis of its mirror; no pennant, as the class is neither $a$ nor $b$ .
6
Cut. 0.42 units between teeth, or half a tooth where teeth are short; square for this class.
7
Field. 168 steps, one for each element of $G$ ; every seventh ends a tooth.

Generator · input to output
\|G:H\|	24	24 teeth
\|H\|	7	7 steps each
covers	7:3	1 ring, 8 × 3
\|N(H):H\|	3	3 blades, hub open
class	fixed	square cut, no pennant

1 · Teeth
2 · Maps
3 · Rotor
4 · Identity

03The family

The fifteen objects

One mark for each conjugacy class of subgroups of $G=\PSL(2,7)$ , by size from 168 down to 1. The sheet is set on its mirror: $a$ -classes to the left of the axis, $b$ -classes to the right, the other nine on it. Fold it along the axis and each $a$ -mark lands on its $b$ -mark.

Read a mark from the outside in: teeth for its points, an inner ring for each object it covers (a pair the mirror exchanges shares one), blades for its automorphisms. The six rigid objects have a gold hub; the names in gold italic are the program’s.

15 marks · 6 rigid · 3 mirror pairs

1168

C_2

C_3

C_4

V_4^a

V_4^b

S_3

28anchored observers

C_7

D_8

A_4^a

A_4^b

7{:}3

8the sky

S_4^a

7clocks

S_4^b

7vantage lines

G

Register of object marks

Each mark with the figures it is generated from: the object’s size $|G:H|$ , the order of its stabilizer, its automorphisms $|N_G(H):H|$ , its conjugates, and its covers: the maps $G/H\to G/K$ with nothing strictly between.

The object of size 168
$G/1$
size
168
stabilizer
1, order 1
automorphisms
168
conjugates
1
covers
$G/C_2$ , $G/C_3$ , $G/C_7$
teeth 168×1 · rings 84×2 + 56×3 + 24×7 · blades 168 · cut square
The object of size 84
$G/C_2$
size
84
stabilizer
$C_2$ , order 2
automorphisms
4
conjugates
21
covers
$G/C_4$ , $G/V_4^a$ , $G/V_4^b$ , $G/S_3$
teeth 84×2 · rings 42×2 + 42×2 pair + 28×3 · blades 4 · cut square
The object of size 56
$G/C_3$
size
56
stabilizer
$C_3$ , order 3
automorphisms
2
conjugates
28
covers
$G/S_3$ , $G/A_4^a$ , $G/A_4^b$ , $G/7{:}3$
teeth 56×3 · rings 28×2 + 14×4 pair + 8×7 · blades 2 · cut square
The object of size 42, cyclic
$G/C_4$
size
42
stabilizer
$C_4$ , order 4
automorphisms
2
conjugates
21
covers
$G/D_8$
teeth 42×4 · rings 21×2 · blades 2 · cut square
The object of size 42, class a
$G/V_4^a$
size
42
stabilizer
$V_4^a$ , order 4
automorphisms
6
conjugates
7
covers
$G/D_8$ , $G/A_4^a$
teeth 42×4 · rings 21×2 + 14×3 a · blades 6 · cut a
The object of size 42, class b
$G/V_4^b$
size
42
stabilizer
$V_4^b$ , order 4
automorphisms
6
conjugates
7
covers
$G/D_8$ , $G/A_4^b$
teeth 42×4 · rings 21×2 + 14×3 b · blades 6 · cut b
The twenty-eightanchored observers
$G/S_3$
size
28
stabilizer
$S_3$ , order 6
automorphisms
1, rigid
conjugates
28
covers
$G/S_4^a$ , $G/S_4^b$
teeth 28×6 · rings 7×4 pair · blades 1 · cut square
The object of size 24
$G/C_7$
size
24
stabilizer
$C_7$ , order 7
automorphisms
3
conjugates
8
covers
$G/7{:}3$
teeth 24×7 · rings 8×3 · blades 3 · cut square
The object of size 21
$G/D_8$
size
21
stabilizer
$D_8$ , order 8
automorphisms
1, rigid
conjugates
21
covers
$G/S_4^a$ , $G/S_4^b$
teeth 21×8 · rings 7×3 pair · blades 1 · cut square
The object of size 14, class a
$G/A_4^a$
size
14
stabilizer
$A_4^a$ , order 12
automorphisms
2
conjugates
7
covers
$G/S_4^a$
teeth 14×12 · rings 7×2 a · blades 2 · cut a
The object of size 14, class b
$G/A_4^b$
size
14
stabilizer
$A_4^b$ , order 12
automorphisms
2
conjugates
7
covers
$G/S_4^b$
teeth 14×12 · rings 7×2 b · blades 2 · cut b
The sky
$G/7{:}3$
size
8
stabilizer
$7{:}3$ , order 21
automorphisms
1, rigid
conjugates
8
covers
$G/G$
teeth 8×21 · rings 1×8 · blades 1 · cut square
The seven pointsclocks
$G/S_4^a$
size
7
stabilizer
$S_4^a$ , order 24
automorphisms
1, rigid
conjugates
7
covers
$G/G$
teeth 7×24 · rings 1×7 · blades 1 · cut a
The seven linesvantage lines
$G/S_4^b$
size
7
stabilizer
$S_4^b$ , order 24
automorphisms
1, rigid
conjugates
7
covers
$G/G$
teeth 7×24 · rings 1×7 · blades 1 · cut b
The object of size 1
$G/G$
size
1
stabilizer
$G$ , order 168
automorphisms
1, rigid
conjugates
1
covers
—
teeth 1×168 · rings — · blades 1 · cut square

04The tower, from above

The thirty-six concepts

A concept’s mark is the plan of the tower seen from above: each floor a ring, the roof outermost, the classical floor at the core, where the objects it is seen on keep their own teeth. Its arc is gold, drawn over the arcs of the concepts it is built from, so every mark contains the arcs of the marks beneath it, stroke for stroke and in place.

Sheet 04.1 · the plan36 sectors · 7 rings

1 L’incarnation

1.1 object
1.2 stabilizer class
1.3 marking
1.4 incarnation
1.5 alignment
1.6 new object

2 Les sutures

2.1 seam
2.2 seam groupoid
2.3 rigid object
2.4 coherence
2.5 seam system
2.6 seam monodromy
2.7 gauge
2.8 power
2.9 quotient class
2.10 twisting element
2.11 seam over an automorphism

3 Ce qui est su

3.1 bridge
3.2 status
3.3 refuted
3.4 description
3.5 kernel

Negative space

3.6 absence
3.7 forced gap
3.8 window
3.9 imprint
3.10 reduction

4 Les doubles vies

4.1 life
4.2 double life
4.3 dictionary

5 Les complétions

5.1 completion
5.2 orientation

6 Les continus

6.1 continuum
6.2 spinor system
6.3 commit algebra

R Le sujet

R seam theory

Sectors are placed by computation: each floor is cut into equal arcs from twelve o’clock and turned to sit nearest the concepts it uses. On floor 3 the negative space keeps one room: the absence and the concepts of its floor built on it.

Floor 0 · Le fonds classique

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.

In the marks, this floor is the core: the rings of the objects a concept is seen on.

Floor 1 · L’incarnation

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.

1.1
object
What is the one thing that several theories name?
A transitive set of a group; up to isomorphism, a conjugacy class of its subgroups.
Arcs
- its own, the first
floor 1 · sector 1/6 · 2°–58° · arcs 1 · core S3 7:3 G
1.2
stabilizer class
What single datum decides which object a set is?
The conjugacy class formed by the stabilizers of an object’s points; it determines the object up to isomorphism.
Arcs of
- object
- and its own
floor 1 · sector 2/6 · 62°–118° · arcs 2 · core S3 S4a S4b
1.3
marking
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.
Arcs of
- object
- stabilizer class
- and its own
floor 1 · sector 3/6 · 122°–178° · arcs 3 · core S3 S4a S4b
1.4
incarnation
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.
Arcs of
- object
- marking
- stabilizer class
- and its own
floor 1 · sector 4/6 · 182°–238° · arcs 4 · core S3 C7 7:3
1.5
alignment
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.
Arcs of
- incarnation
- object
- and its own
- $S_3$
- $C_7$
floor 1 · sector 5/6 · 242°–298° · arcs 5 · core S3 C7
1.6
new object
Which objects does a double cover add to those of the group below it?
A 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.
Arcs of
- object
- stabilizer class
- incarnation
- and its own
floor 1 · sector 6/6 · 302°–358° · arcs 5 · core 1 C3 C7 7:3

Floor 2 · Les sutures

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.

2.1
seam
How are two incarnations of one object matched?
An equivariant bijection between two incarnations of one object; the seams between two incarnations form a torsor under the object’s automorphisms.
Arcs of
- incarnation
- alignment
- and its own
- $S_3$
floor 2 · sector 7/11 · 198°–227° · arcs 6 · core S3
2.2
seam groupoid
How are all the seams of a family recorded at once?
The groupoid whose vertices are a family of incarnations and whose arrows are their seams; consistency means it is the pair groupoid.
Arcs of
- seam
- and its own
- $S_3$
- $C_7$
floor 2 · sector 11/11 · 329°–358° · arcs 7 · core S3 C7
2.3
rigid object
When is the seam between two incarnations forced?
An object with no automorphism but the identity; equivalently its stabilizers are self-normalizing, and then every seam is unique.
Arcs of
- seam
- alignment
- stabilizer class
- and its own
floor 2 · sector 10/11 · 297°–325° · arcs 7 · core S3 D8 7:3 S4a S4b G
2.4
coherence
Do seams chosen one at a time agree around every route?
A 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.
Arcs of
- seam groupoid
- rigid object
- alignment
- and its own
floor 2 · sector 6/11 · 166°–194° · arcs 9 · core C3 S3 C7
2.5
seam system
Which seams do the theories themselves supply?
A family of incarnations with a chosen set of seams among them, loops allowed: typically the natural identifications that the theories provide.
Arcs of
- seam
- incarnation
- and its own
floor 2 · sector 8/11 · 231°–260° · arcs 7 · core C3 C4 C7 S4a
2.6
seam monodromy
What does going around a loop of natural identifications do?
The 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.
Arcs of
- seam system
- coherence
- alignment
- and its own
floor 2 · sector 1/11 · 2°–31° · arcs 11 · core C4 V4b C7
2.7
gauge
What does a seam system become once an alignment is chosen at every incarnation?
A 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.
Arcs of
- seam system
- seam monodromy
- alignment
- and its own
- $S_3$
- $C_7$
floor 2 · sector 2/11 · 35°–63° · arcs 12 · core S3 C7
2.8
power
How can automorphisms of incarnations in different theories be compared?
For 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.
Arcs of
- seam
- incarnation
- and its own
floor 2 · sector 9/11 · 264°–292° · arcs 7 · core C3 C4 C7
2.9
quotient class
How are automorphisms named when no power is available?
The 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.
Arcs of
- object
- stabilizer class
- seam
- and its own
- $C_2$
- $C_7$
floor 2 · sector 3/11 · 68°–96° · arcs 7 · core C2 C7
2.10
twisting element
What does a symmetry that normalizes the group, rather than commuting with it, give?
The 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.
Arcs of
- marking
- power
- and its own
- $C_7$
floor 2 · sector 4/11 · 100°–129° · arcs 8 · core C7
2.11
seam over an automorphism
What is a seam after twisting by an automorphism of the group?
A 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.
Arcs of
- seam
- marking
- rigid object
- and its own
floor 2 · sector 5/11 · 133°–162° · arcs 8 · core S3 S4a S4b

Floor 3 · Ce qui est su

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.

3.1
bridge
What exactly is claimed when two theories are said to name the same thing?
The assertion that two sets, given in two theories, are incarnations of one object; only a built bridge is a theorem.
Arcs of
- incarnation
- seam
- and its own
floor 3 · sector 7/10 · 218°–250° · arcs 7 · core A4b 7:3 S4a
3.2
status
How much is known about a bridge?
Built, type, name or refuted: a record of what is known about a bridge, not a property of the objects.
Arcs of
- bridge
- and its own
floor 3 · sector 8/10 · 254°–286° · arcs 8 · core S3 A4a A4b
3.3
refuted
When is it proved that two sets are not one object?
The status of a bridge proved false: no seam exists for the markings in question. A type recurrence proved to be no identification.
Arcs of
- bridge
- status
- marking
- stabilizer class
- and its own
floor 3 · sector 6/10 · 182°–214° · arcs 9 · core V4a V4b A4a A4b S4a S4b
3.4
description
What is a map between theories that is not a seam?
A 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.
Arcs of
- object
- seam
- and its own
floor 3 · sector 9/10 · 290°–322° · arcs 7 · core S3 S4a S4b
3.5
kernel
What does a description forget?
The 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.
Arcs of
- description
- and its own
floor 3 · sector 10/10 · 326°–358° · arcs 8 · core S3 C7 7:3
3.6
absence
What is a theorem that something does not exist, taken as an object of study?
A 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.
Arcs of
- refuted
- and its own
floor 3 · sector 4/10 · 110°–142° · arcs 10 · core 1 C2 V4a V4b A4a A4b
3.7
forced gap
Which objects can the simplest figures of a theory not reach?
A 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.
Arcs of
- absence
- incarnation
- stabilizer class
- and its own
floor 3 · sector 3/10 · 74°–106° · arcs 11 · core 1 V4a V4b C7
3.8
window
Where can the excluded thing still happen?
For a graded absence, the set of parameter values that actually occur; it is usually small, and its edges carry the structure.
Arcs of
- absence
- and its own
- $S_4^a$
- $S_4^b$
floor 3 · sector 1/10 · 2°–34° · arcs 11 · core S4a S4b
3.9
imprint
What structure does an absence force to exist?
A 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).
Arcs of
- absence
- window
- and its own
floor 3 · sector 2/10 · 38°–70° · arcs 12 · core V4a V4b A4a A4b S4b
3.10
reduction
Which absences are the same fact seen twice?
An 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.
Arcs of
- absence
- imprint
- bridge
- and its own
floor 3 · sector 5/10 · 146°–178° · arcs 13 · core 7:3 S4a S4b

Floor 4 · Les doubles vies

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.

4.1
life
In which classical geometry does a group live?
An 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.
Arcs of
- marking
- object
- and its own
floor 4 · sector 1/3 · 2°–118° · arcs 4 · core 7:3 S4a S4b
4.2
double life
Which groups carry the geometries of two families at once?
Two 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 $A_8$ .
Arcs of
- life
- absence
- marking
- and its own
floor 4 · sector 2/3 · 122°–238° · arcs 12 · core S3 7:3 S4a S4b
4.3
dictionary
Which natural sets of one life are which natural sets of the other?
For each object of a group with a double life, the natural sets of each life that are incarnations of it, computed by matching stabilizers.
Arcs of
- double life
- stabilizer class
- incarnation
- and its own
floor 4 · sector 3/3 · 242°–358° · arcs 13 · core C3 S3 C7 D8 A4a A4b 7:3 S4a

Floor 5 · Les complétions

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.

5.1
completion
Where does a finite geometry sit inside a building over a local field?
A 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.
Arcs of
- double life
- life
- incarnation
- and its own
floor 5 · sector 2/2 · 181°–359° · arcs 13 · core S3 D8 7:3 S4a S4b
5.2
orientation
What does each parent’s flip change, and what can see it?
Each 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.
Arcs of
- seam over an automorphism
- completion
- gauge
- and its own
floor 5 · sector 1/2 · 1°–179° · arcs 21 · core A4a A4b 7:3

Floor 6 · Les continus

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.

6.1
continuum
How does a finite object reappear inside a continuous geometry?
A homogeneous space of a Lie group that carries the object, either as an equivariant configuration (embedded) or as classes modulo a congruence subgroup (arithmetic).
Arcs of
- object
- completion
- absence
- and its own
floor 6 · sector 3/3 · 241°–359° · arcs 14 · core C3 S3 A4a A4b 7:3
6.2
spinor system
What does the defining representation of the Bianchi group carry at the cusps of the link complement?
The 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.
Arcs of
- continuum
- orientation
- and its own
- $7{:}3$
- $S_4^a$
floor 6 · sector 1/3 · 1°–119° · arcs 23 · core 7:3 S4a
6.3
commit algebra
Which operators commute with every move at a point of the Fano plane?
At 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.
Arcs of
- spinor system
- orientation
- and its own
- $S_4^a$
floor 6 · sector 2/3 · 121°–239° · arcs 24 · core S4a

The roof · Le sujet

What is studied?

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.

R
seam theory
What is the subject?
The 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.
Arcs of
- stabilizer class
- seam
- rigid object
- seam monodromy
- marking
- kernel
- absence
- imprint
- and its own
floor roof · sector 1/1 · 1°–359° · arcs 20 · core S3 C7 7:3 S4a S4b

05In depth · 2.6

Seam monodromy

What does going around a loop of natural identifications do?

The 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.

Sheet 05.1 · the mark of seam monodromy, every arc namedThe entry

its own2.6 seam monodromy
beneath1.1 object
beneath1.2 stabilizer class
beneath1.3 marking
beneath1.4 incarnation
uses1.5 alignment
beneath2.1 seam
beneath2.2 seam groupoid
beneath2.3 rigid object
uses2.4 coherence
uses2.5 seam system

Gold, its own arc; ink, the three concepts it uses; pale, what those are built from. At the core, the teeth of the three objects it is seen on, outermost first:

G/C_4

, 42 cut square;

G/V_4^b

, 42 sheared to the right; and

G/C_7

, 24 cut square.

DefinitionSeam system, monodromy

The monodromy of a cycle $\gamma=(s_1^{\epsilon_1},\dots,s_m^{\epsilon_m})$ of a seam system, starting and ending at $Y$ , is

\mathrm{mon}(\gamma)=s_m^{\epsilon_m}\circ\cdots\circ s_1^{\epsilon_1}\in\Aut_G(Y).

The word is used as for coverings: going around a loop of identifications returns a permutation of the fibre. Here the fibre is an incarnation and the permutation is an automorphism of the object; through an alignment it is an element of $N_G(H)/H$ , well defined up to conjugation.

Proposition

If $N_G(H)/H$ is abelian, then for each incarnation $Y$ the isomorphism $\Aut_G(Y)\cong\Aut_G(X)$ given by an alignment does not depend on the alignment, and monodromy is a homomorphism from the fundamental group of the graph of the system to $\Aut_G(X)\cong N_G(H)/H$ .

Proof

Two alignments differ by an automorphism $a$ of $X$ , and the two isomorphisms differ by conjugation by $a$ , which is trivial in an abelian group. Concatenating cycles composes monodromies.

TheoremMonodromy of the object of size 24

Let $\tau$ be the composite of the tangent and residual-point seams: a flex of the Klein quartic goes to the other flex on its tangent. Then $\tau$ is an automorphism of the flexes of power 4, so the cycle flexes $\to$ flex tangents $\to$ flexes, along the two natural seams, has monodromy of order 3. It permutes each flex triangle cyclically:

\tau\colon\ (1:0:0)\mapsto(0:0:1)\mapsto(0:1:0)\mapsto(1:0:0).

Proof

The rotation of $(0:0:1)$ is $g$ , and the rotation of $(0:1:0)$ is the element acting there by $\zeta$ , which is $g^4$ , since $\rho(g)$ acts there by $\zeta^2$ and so $\rho(g)^4$ by $\zeta^8=\zeta$ . As $\tau(0:0:1)=(0:1:0)$ , the rotation seam carries $\tau$ to a map sending $g$ to $g^4$ , which is the fourth-power map.

Sheet 05.2 · the loop, in marksthe object of size 24

00 At rest: the gold blade at 1, the identity; the flex token at (1:0:0).

The monodromy

\tau

on one flex triangle of the Klein quartic: each flex goes along its tangent to the other flex on it, and three steps return. Both ends of the loop are incarnations of one object, so both wear the mark of

G/C_7

; the dial around its rotor reads the powers 1, 2 and 4.

Remark

So the answer for non-rigid objects is mixed. Seams fixed by the conventions of their theories are consistent wherever they meet. But a single theory may supply two natural seams between the same two incarnations, and then a cycle of length two already has nontrivial monodromy: the contact point and the residual point of a flex tangent, or the roles of a point in its line, whose three seams have relative powers 1, 2 and 4. The monodromy is then an invariant of the theory; here it is the cyclic order that the tangents put on each flex triangle, a fact of the projective geometry of the quartic.

Sheet 05.3 · the three role seams

The role seams from the cyclic labellings for

\{0,1,3\}

to the perfect matchings of the Heawood graph. Relative to the role 0, the roles 1 and 3 have powers 2 and 4: each lands on the blade of its power.

TheoremThe Coxeter edges and the marking

Let the Coxeter graph be in its antiflag model, with $G$ acting through a marking $\mu$ . Each edge has the form $\{(p,B),(q,B')\}$ , with $B$ and $B'$ meeting in the third point $c$ of the line $pq$ .

(a) The point rule, which goes from each point of $B$ off $pq$ to the third point of its line with $p$ , and from each point of $B'$ off $pq$ to the third point of its line with $q$ , traces a directed 4-cycle on the quadrangle complementary to $pq$ . The line rule traces, dually, a directed 4-cycle on the four lines missing $c$ . Each rule, followed by the element of order 4 that advances its cycle one step, is a seam from the edges to $4A$ , and the two rules give mutually inverse elements.

(b) The vertex seam sends an antiflag to the pair of points of $\Proj^1(\F_7)$ with the same stabilizer, and an edge to a harmonic pair of disjoint pairs $\{\{a,b\},\{c,d\}\}$ . Of the two directed 4-cycles $a\to c\to b\to d\to a$ and $a\to d\to b\to c\to a$ , exactly one has $[a,c][c,b][b,a]$ a nonzero square, and the bracket rule sends the edge to the element of order 4 advancing that cycle one step.

(c) If $\mu$ differs from $\mu_A$ by an inner automorphism, the bracket rule agrees with the point rule on every edge; if by an outer one, it agrees with the line rule.

Consequently the seam system for $G/C_4$ formed by the Coxeter edges, the harmonic pairs of pairs and the class $4A$ , with the vertex seam, the bracket rule and the point rule, is coherent when the marking is in the class of $\mu_A$ , and its monodromy is the nontrivial automorphism otherwise.

Proof

(a) The rules use only incidence and treat the two antiflags of an edge alike, so they are $G$ -maps; that they give inverse elements was checked by machine. (b) In $[a,c][c,b][b,a]$ each point occurs twice, so its square class does not depend on the coordinate vectors, and it is invariant under $\SL(2,7)$ . With $a=0$ , $b=\infty$ , harmonicity gives $d=-c$ , and the products for the cycle $a\to c\to b\to d$ are all in the square class of $c$ , while the reverse cycle gives that of $-c$ ; as $-1$ is not a square modulo 7, exactly one cycle has a square product. (c) For $\mu_A$ the agreement was checked on all 42 edges. An inner change of marking is induced by a collineation, which commutes with all the constructions. An outer change, by conjugation with a Möbius map of non-square determinant, multiplies every bracket by a non-square, so it reverses the bracket rule.

PropositionMonodromy as the kernel of holonomy

Let a seam system over a connected graph $\mathcal G$ have holonomy $\mathrm{hol}\colon\pi_1(\mathcal G,v)\to A$ , read as a description; its kernel $N$ is the group of loops around which the seams close up. The system is coherent if and only if $N=\pi_1(\mathcal G,v)$ , and the group of monodromies is $\pi_1(\mathcal G,v)/N$ . So monodromy is what remains of the loops once the kernel of the holonomy is divided out.

Proof

Holonomy is a homomorphism on the fundamental group, and the system is coherent exactly when every holonomy is trivial.

Examplecomputed

Monodromy can be the spinor sign. In the lattice $E_8$ preserved by $\SL(2,7)$ for one class of tetrahedra of Thurston’s manifold, the 224 half-roots form two copies $O_1$ and $O_2$ of the new object of size 112, whose automorphism group is $C_4$ . The reflection seam, changing the sign of a half-root at the point of its support fixed by its stabilizer, is a seam from $O_1$ to $O_2$ and back, and the cycle it forms has trivial monodromy, since its square is the identity. The sign seam, the sign pattern of $Cr$ on the support of $r$ , with $C$ the conference matrix of the Weil representation, equals the reflection seam on $O_1$ and its negative on $O_2$ : the cycle it forms has monodromy $-I$ . Half of $Cr$ off the support is an automorphism of $O_2$ of order 4 with square $-1$ , the integral shadow of multiplication by $i$ , and it generates the automorphism group.

Open questionopen

On the object $G/V_4^b$ the automorphism group is $S_3$ , not abelian, so monodromy is defined only up to conjugation. A natural seam system with non-abelian monodromy is not known: the natural seams found there, the elation and centre seams, carry swaps to swaps and rotations to rotations, and are coherent.

The mark of

G/V_4^b

: six blades, for an automorphism group

S_3

that is not abelian. Here a turn of the gold blade is defined only up to conjugation.

06Colophon

Editions

A generative identity is usually reseeded: each printing draws new marks. This one has no seed to change. Every mark is computed from the figures below, which come from lib/math/subgroups and the concepts’ floors, uses and objects. Reprint it, and every tooth falls where it fell. The fingerprint is taken over all of the inputs; it moves only if the mathematics does.

It has moved twice. The first edition, 9E241517, was printed from the first draft of Seams: fifteen objects, twenty-seven concepts and five questions. The second, FEBD7C3E, from the revised draft: thirty-three concepts and six questions. This, the third, has thirty-six concepts and six questions; orientation, spinor system and commit algebra are new. The object inputs have never changed, and neither have the fifteen object marks.

Against the second edition, the arcs of completion and continuum moved to make room; every other arc is where it was. Every concept mark is redrawn all the same, because the dotted plan beneath each one gained three sectors.

FingerprintA90921DD

objects: F27A86A3
concepts: EDF4C773
first edition: 9E241517
second edition: FEBD7C3E

Object inputs
class	\|G:H\|	\|H\|	aut	covers, index
1	168	1	168	C2 2, C3 3, C7 7
C2	84	2	4	C4 2, V4a+V4b 2, S3 3
C3	56	3	2	S3 2, A4a+A4b 4, 7:3 7
C4	42	4	2	D8 2
V4a	42	4	6	D8 2, A4a 3
V4b	42	4	6	D8 2, A4b 3
S3	28	6	1	S4a+S4b 4
C7	24	7	3	7:3 3
D8	21	8	1	S4a+S4b 3
A4a	14	12	2	S4a 2
A4b	14	12	2	S4b 2
7:3	8	21	1	G 8
S4a	7	24	1	G 7
S4b	7	24	1	G 7
G	1	168	1	—

Concept inputs
code	concept	uses	objects
1.1	object	—	S3 7:3 G
1.2	stabilizer-class	1.1	S3 S4a S4b
1.3	marking	1.1 1.2	S3 S4a S4b
1.4	incarnation	1.1 1.3 1.2	S3 C7 7:3
1.5	alignment	1.4 1.1	S3 C7
1.6	new-object	1.1 1.2 1.4	1 C3 C7 7:3
2.1	seam	1.4 1.5	S3
2.2	seam-groupoid	2.1	S3 C7
2.3	rigid-object	2.1 1.5 1.2	S3 D8 7:3 S4a S4b G
2.4	coherence	2.2 2.3 1.5	C3 S3 C7
2.5	seam-system	2.1 1.4	C3 C4 C7 S4a
2.6	seam-monodromy	2.5 2.4 1.5	C4 V4b C7
2.7	gauge	2.5 2.6 1.5	S3 C7
2.8	power	2.1 1.4	C3 C4 C7
2.9	quotient-class	1.1 1.2 2.1	C2 C7
2.10	twisting-element	1.3 2.8	C7
2.11	seam-over-automorphism	2.1 1.3 2.3	S3 S4a S4b
3.1	bridge	1.4 2.1	A4b 7:3 S4a
3.2	status	3.1	S3 A4a A4b
3.3	refuted	3.1 3.2 1.3 1.2	V4a V4b A4a A4b S4a S4b
3.4	description	1.1 2.1	S3 S4a S4b
3.5	kernel	3.4	S3 C7 7:3
3.6	absence	3.3	1 C2 V4a V4b A4a A4b
3.7	forced-gap	3.6 1.4 1.2	1 V4a V4b C7
3.8	window	3.6	S4a S4b
3.9	imprint	3.6 3.8	V4a V4b A4a A4b S4b
3.10	reduction	3.6 3.9 3.1	7:3 S4a S4b
4.1	life	1.3 1.1	7:3 S4a S4b
4.2	double-life	4.1 3.6 1.3	S3 7:3 S4a S4b
4.3	dictionary	4.2 1.2 1.4	C3 S3 C7 D8 A4a A4b 7:3 S4a
5.1	completion	4.2 4.1 1.4	S3 D8 7:3 S4a S4b
5.2	orientation	2.11 5.1 2.7	A4a A4b 7:3
6.1	continuum	1.1 5.1 3.6	C3 S3 A4a A4b 7:3
6.2	spinor-system	6.1 5.2	7:3 S4a
6.3	commit-algebra	6.2 5.2	S4a
R	seam-theory	1.2 2.1 2.3 2.6 1.3 3.5 3.6 3.9	S3 C7 7:3 S4a S4b

SeamTheory

Existence

Teeth

Number

Rotor

Consistency

Gold

Dependence on markings

Mirror

Forgetting

Rings

Impossibility

Pairs

Register of object marks

Seam
Theory