Universal Kernel · a zine

SeamTheory

a zine about the joins, not the pieces

worked through, floor by floor: the group of order 168

Hi, I’m Seamus. I sew. The boxes do the maths.

Gist

The roof: le sujet

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.

the roof asks

Six questions

Seams states its subject as six questions. Each panel quotes one and points to the concepts that answer it, further up this zine.

Existence
two sets are joined exactly when their stabilizer classes agree.
answered bystabilizer class · floor 1
Number
the seams between two incarnations form a torsor under $N_G(H)/H$ .
answered byseam · floor 2
Consistency
seams are unique and coherent exactly when the object is rigid, and otherwise natural seams can carry monodromy.
answered byrigid object · floor 2 seam monodromy · floor 2
Dependence on markings
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.
answered bymarking · floor 1
Forgetting
a map between theories that is not a seam is a description, and what it forgets at a point is its kernel, a stabilizer.
answered bydescription · floor 3 kernel · floor 3
Impossibility
the negative space of absences, with their windows and imprints.
answered byabsence · floor 3 imprint · floor 3

cut along the dotted line, then climb

36 concepts, floor by floor

The tower

An arrow from A to B means B is built from A. The map inks the arrows floor by floor; point at a concept, on the map or on its card, to see everything it is built from.

Floor 0

Le fonds classique

Q. What does each theory supply before anything is compared?

Gist

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.

Floor 1

L’incarnation

Q. When do two theories name one object?

Gist

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.

built fromnothing before it: the tower starts here

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.

built fromobject

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.

built fromobject stabilizer class

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.

built fromobject marking stabilizer class

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.

built fromincarnation object

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.

built fromobject stabilizer class incarnation

Floor 2

Les sutures

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

Gist

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.

built fromincarnation alignment

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.

built fromseam

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.

built fromseam alignment stabilizer class

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.

built fromseam groupoid rigid object alignment

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.

built fromseam incarnation

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.

built fromseam system coherence alignment

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.

built fromseam system seam monodromy alignment

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.

built fromseam incarnation

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.

built fromobject stabilizer class seam

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.

built frommarking power

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.

built fromseam marking rigid object

Floor 3

Ce qui est su

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

Gist

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.

built fromincarnation seam

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.

built frombridge

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.

built frombridge status marking stabilizer class

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.

built fromobject seam

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.

built fromdescription

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.

built fromrefuted

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.

built fromabsence incarnation stabilizer class

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.

built fromabsence

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).

built fromabsence window

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.

built fromabsence imprint bridge

Floor 4

Les doubles vies

Q. When does one group carry the geometries of two families?

Gist

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.

built frommarking object

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 $A_8$ .

built fromlife absence marking

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.

built fromdouble life stabilizer class incarnation

Floor 5

Les complétions

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

Gist

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.

built fromdouble life life incarnation

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.

built fromseam over an automorphism completion gauge

Floor 6

Les continus

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

Gist

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).

built fromobject completion absence

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.

built fromcontinuum orientation

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.

built fromspinor system orientation

The roof

Le sujet

Q. What is studied?

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.

built fromstabilizer class seam rigid object seam monodromy marking kernel absence imprint

Remark

The tower assembled

Each floor is joined to the one below it by a theorem. The stabilizer principle joins incarnation to the classical floor. Seams join incarnations, unique when the stabilizer is self-normalizing and otherwise carrying monodromy. A bridge has a status, which the stabilizer principle decides for two marked sets of one group, and beside the statuses stand the absences. Artin’s absence leaves exactly four groups with a double life. Residue fields make each life the link of a vertex of a building over a local field: at 2 the octonion table glues such links into a building over $\F_2(\!(t)\!)$ and Kato’s hermitian form glues them, without symmetry, into the building over $\Q_2$ ; a gluing with the Frobenius symmetry of order 21 is the octonion one; and over $\Z[1/14]$ the group of order 168 is the stabilizer of the vertex of Klein’s lattice, carrying both lives, where the finite geometry is the geometry of short vectors and neighbours; one step beyond the link the two trees at 7 carry a doublet and its symmetric square, the object of the points is carried along one tree and not the other, and the two parents carry independent flips, of which only the first is seen by the oriented cells of the link complement. The two parents are joined only by fiber products, which across scales keep no attachment of the finite line and at one scale force it; around the loops of the scale tree a single relabelling carries the signed table without reversals exactly on the Iwahori subgroup, and on seven loops in eight must reverse two units, while carried observer by observer every loop returns each fiber changed only by colour. The archimedean place gives, over $\C$ , Thurston’s link complement, whose cells are objects of the group, with the Fano incidence among them the absence of a shared face; the group has two arithmetic parents at 7, which share the sky but not its completion; absences decide which continua exist; and in the Cayley plane one point and one imaginary unit carry the intersection of Todorov and Dubois-Violette.

the fifteen objects, please step forward

the fifteen objects

The line-up

One portrait for each object of the group of order 168, largest first: a dot for each of its elements. Each portrait links to its entry.

1168168 seams

C_2

844 seams

C_3

562 seams

C_4

422 seams

V_4^a

426 seams

V_4^b

426 seams

S_3

28rigid1 seamanchored observers

C_7

243 seams

D_8

21rigid1 seam

A_4^a

142 seams

A_4^b

142 seams

7{:}3

8rigid1 seamthe sky

S_4^a

7rigid1 seamclocks

S_4^b

7rigid1 seamvantage lines

G

1rigid1 seam

ruled with a ruler: rigid
drawn freehand: not rigid
3 seams how many seams join any two of its incarnations
lookalikes same permutation character, different objects
the sky the program’s own name, where it has one

Examplecomputed

The group $G=\PSL(2,7)$ of order 168 has exactly 179 subgroups, in fifteen conjugacy classes, in agreement with Dickson’s classification. So it has exactly fifteen objects, of sizes

168,\ 84,\ 56,\ 42,\ 42,\ 42,\ 28,\ 24,\ 21,\ 14,\ 14,\ 8,\ 7,\ 7,\ 1,

with stabilizers 1, $C_2$ , $C_3$ , $C_4$ , $V_4^a$ , $V_4^b$ , $S_3$ , $C_7$ , $D_8$ , $A_4^a$ , $A_4^b$ , $7{:}3$ , $S_4^a$ , $S_4^b$ and $G$ . The labels $a$ and $b$ are fixed by a marking: $S_4^a$ is the class of the stabilizers of the points of the Fano plane, $S_4^b$ that of its lines, and $V_4^a$ , $A_4^a$ lie in a member of $S_4^a$ as its normal Klein four-group and its alternating group.

Corollarycomputed

Rigid and non-rigid objects

Exactly six of the fifteen objects of $\PSL(2,7)$ are rigid: those with stabilizers $S_3$ , $D_8$ , $7{:}3$ , $S_4^a$ , $S_4^b$ and $G$ , of sizes 28, 21, 8, 7, 7 and 1. For each of them all seams between incarnations are unique and consistent. The other nine, of sizes 168, 84, 56, 42, 42, 42, 24, 14 and 14, have automorphism groups $G$ , $C_2\times C_2$ , $C_2$ , $C_2$ , $S_3$ , $S_3$ , $C_3$ , $C_2$ and $C_2$ .

Theorem

The stabilizer principle

Let $X$ be an object with stabilizer class $[H]$ . Then $\Aut_G(X)\cong N_G(H)/H$ , and for two incarnations $Y$ and $Y'$ the seams $Y\to Y'$ form a torsor for $\Aut_G(Y)$ acting by precomposition and for $\Aut_G(Y')$ acting by postcomposition. There are exactly $|N_G(H):H|$ of them, and one is written down as soon as a pair of points with equal stabilizers is found: $gy\mapsto gy'$ .

Unfold the proofProof

If $s,s'$ are seams, $s^{-1}s'$ is an automorphism of $Y$ and $s'=s\circ(s^{-1}s')$ . The automorphisms of $X=Gx$ correspond to the points $nx$ with $n\in N_G(H)$ , and $nx=x$ exactly when $n\in H$ .

Proposition

The Gassmann pairs of PSL(2,7)

Two distinct objects of $\PSL(2,7)$ have the same permutation character exactly for the three pairs $(G/V_4^a,G/V_4^b)$ , $(G/A_4^a,G/A_4^b)$ and $(G/S_4^a,G/S_4^b)$ . Every bridge between the two members of such a pair, for one marking, is refuted.

Unfold the proofProof

The outer automorphism exchanges the two members of each pair and fixes every conjugacy class of elements except $7A$ and $7B$ . The groups $V_4$ , $A_4$ and $S_4$ contain no element of order 7, so the two members meet each conjugacy class equally often, which is the condition for equal permutation characters. Their marks at $V_4^a$ differ, so by Burnside’s theorem they are not isomorphic $G$ -sets, and no seam joins them.

one concept, slowly

in depth · floor 2 · entry 2.6

Seam monodromy

Q. 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.

built fromseam system coherence alignment the full entry ↗

Definition

Seam 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.

The loop

out along the tangent, back by the residual point

Round 1

Here we go.

tangent

(1:0:0)\mapsto\{y=0\}

residual point

\{y=0\}\mapsto(0:0:1)

The first time round the loop.

Round 2

Again!

tangent

(0:0:1)\mapsto\{x=0\}

residual point

\{x=0\}\mapsto(0:1:0)

The second time round the loop.

Round 3

And once more.

tangent

(0:1:0)\mapsto\{z=0\}

residual point

\{z=0\}\mapsto(1:0:0)

The third time round the loop.

Theorem

Monodromy 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).

Unfold the proofProof

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.

What comes back

the powers

K_{C_7}=\{1,2,4\}

\tau

power 4

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 $\Aut_G(X)\cong N_G(H)/H$ .

Unfold the proofProof

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.

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.

Theorem

The 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.

Unfold the proofProof

(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.

Proposition

Monodromy 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.

Unfold the proofProof

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.

SeamTheory

The roof: le sujet

Existence

Number

Consistency

Dependence on markings

Forgetting

Impossibility

The tower assembled

Rigid and non-rigid objects

The stabilizer principle

The Gassmann pairs of PSL(2,7)

Seam monodromy

Seam system, monodromy

Monodromy of the object of size 24

The Coxeter edges and the marking

Monodromy as the kernel of holonomy