absence

Floor 3, Ce qui est su · introduced in Chapter 2, L’espace en creux

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.

Plate 3.6The negative space of three kinds of figure. Gold dots are incarnations of the fifteen objects; blue hatching marks the cells that are empty by necessity.

Definition(Absence, window, witness)

An absence is a theorem asserting that a collection $\mathcal K$ of structures, specified by explicit axioms, has no member with a stated property. A witness of an absence is a finite configuration on which the absence can already be checked.

The word is preferred to non-existence theorem because what interests the book is the shape the theorem gives to the mathematics around it, not the bare negation. An exceptional isomorphism lets two families of groups touch, and the theorem that there are no others says the touching is isolated.

Proposition(Forced gaps)

(a) A conjugacy class of elements of $G=\PSL(2,7)$ is an incarnation of $G/C_G(x)$ , and the centralizers are $G$ , $D_8$ , $C_3$ , $C_4$ , $C_7$ , $C_7$ ; a conjugacy class of subgroups is an incarnation of $G/N_G(H)$ , and the normalizers are $G$ , $D_8$ , $S_3$ , $7{:}3$ , $S_4^a$ , $S_4^b$ . So the objects with stabilizers 1, $C_2$ , $V_4^a$ , $V_4^b$ , $A_4^a$ , $A_4^b$ are incarnated by no conjugacy class of elements or of subgroups.

(b) No point of $\Proj^2(\C)$ has stabilizer, under Klein’s representation, in the classes $V_4^a$ , $V_4^b$ , $A_4^a$ , $A_4^b$ , $7{:}3$ , $S_4^a$ , $S_4^b$ or $G$ , and the same holds for lines.

Proof

(a) The stabilizer of an element under conjugation is its centralizer, and that of a subgroup its normalizer.

(b) A point fixed by $K$ is a line of $\C^3$ invariant under $\rho(K)$ . The restrictions of $\rho$ to $A_4$ , $S_4$ and $7{:}3$ have norm 1, so they are irreducible and fix no point. The involutions of a Klein four-group have eigenvalues $1,-1,-1$ and are simultaneously diagonal, so its only fixed points are the centres of its three involutions, and the stabilizer of the centre of $a$ is the centralizer of $a$ , a $D_8$ . For lines, apply the same argument to the contragredient representation, which is $\rho$ composed with an outer automorphism.

(c) This is part of Elkies’s description of the orbits on the quartic: the orbits with nontrivial stabilizer have 24, 56 and 84 points, with stabilizers $C_7$ , $C_3$ and $C_2$ .

Remark(The catalogue)

The book’s catalogue has eleven absences, each with its window, its imprint and a witness.

$\PSL(2,p)$ has no subgroup of index $p$ for $p>11$ : window $\{5,7,11\}$ ; imprints $\Proj^1(\F_4)$ with $A_5$ , the Fano plane and the 11-point biplane.

$\SL(2,p)$ has no nontrivial two-dimensional representation for $p\ge7$ : window $\{3,5\}$ ; imprints $2T$ and $2I$ , and the spin bundle, with $\operatorname{Ind}\rho_\pm=6_\pm\oplus8$ .

$\PSL(2,7)$ has no faithful real representation of dimension less than 6: imprint the complex structure of the realification of 3, with commutant $\C$ .

$\PSL(3,q)\not\cong\PSL(2,r)$ unless $(q,r)=(2,7)$ : imprint the group of order 168 with its two lives.

$\PSL(3,4)\not\cong A_8$ : imprint the element orders, and two $M_{24}$ -sets of size 12144; witness $(1\,2\,3\,4\,5)(6\,7\,8)$ , of order 15.

Fano’s axiom fails only in characteristic 2: window $\{2\}$ ; imprint the diagonal line, $\mathrm{PG}(2,2)$ as $K_4$ completed by its matchings.

No compact Riemann surface of genus $g\ge2$ has more than $84(g-1)$ automorphisms: imprint $\Delta(2,3,7)$ and the Klein quartic.

There is no composition algebra outside dimensions 1, 2, 4, 8: imprint $\Oct$ ; witness $(e_1+e_{10})(e_4-e_{15})=0$ in the sedenions.

There is no octonionic projective space of dimension at least 3, and $\mathfrak h_n(\Oct)$ is not a Jordan algebra for $n\ge4$ : imprints $\Oct\mathrm P^2$ with $F_4$ , and $\mathfrak h_3(\Oct)$ .

In dimension $d\ge3$ every frame function is of trace form: the window of frame functions not of trace form is $\{2\}$ ; imprints the density operator, and the effects at $d=2$ .

In dimension $d\ge3$ there is no valuation: window $\{1,2\}$ ; imprints the qubit, and the density operator as carrier; witness Peres’s 33 rays.

Examplecomputed

Small absences are recorded as data too. In the Coxeter graph the cycles of lengths 7, 8, 9 and 10 form one orbit each, with stabilizer classes $C_7$ , $D_8$ , $C_3$ and $C_2$ ; there are no cycles of length 11; and the cycles of length 12 form two orbits, of classes $C_3$ and $C_2$ .

Proposition(Absences as empty fibres)

Let $Z$ be a finite $G$ -set and let $\tau$ send $z\in Z$ to the conjugacy class of $G_z$ , its orbit type. The fibre of $\tau$ over the class of $H$ has $|G:H|\,c_H(Z)$ points, where $c_H(Z)$ is the number of orbits of $Z$ isomorphic to $G/H$ , and the numbers $c_H(Z)$ are obtained from the marks $|Z^K|$ by inverting the table of marks. So a forced gap of a family of figures is an empty fibre of the orbit-type description, and it is detected by the marks. In the same way the window of an absence is the image of its invariant: the values whose fibre is not empty.

Proof

$Z$ is the disjoint union of $c_H(Z)$ copies of $G/H$ over the classes, so $|Z^K|=\sum_Hc_H(Z)\,m(H,K)$ , and the matrix of marks is invertible.

Example(Incidence as an absence)

The Fano incidence appears in the cells of a hyperbolic manifold as an absence. In Thurston’s congruence link complement, call the complementary pairs of tetrahedra of class $a$ points and those of class $b$ lines. A point lies on a line exactly when no tetrahedron of the one shares a face with a tetrahedron of the other, and this incidence is the Fano plane. The reason is a self-dual code: the tetrahedra of one class are the blocks of a Steiner system $S(3,4,8)$ on the eight cusps, and having no shared face is orthogonality in its code.

Remark(No action on four-dimensional space)

The group of order 168 has no nontrivial homomorphism into $\GL(4,\R)$ : it is simple, so a nontrivial homomorphism is injective, and it has no faithful real representation of dimension less than 6. So it is a subgroup neither of the Lorentz group of $\R^{1,3}$ nor of $\PSL(2,\C)$ , and it can meet $\PSL(2,\C)$ only as a quotient of a subgroup, as it does for the arithmetic group $\PSL(2,\Z[\zeta])$ .

Built from: refuted
Builds: forced gapwindowimprintreductiondouble lifecontinuumseam theory
Objects: the object of size 168the object of size 84the object of size 42, class athe object of size 42, class bthe object of size 14, class athe object of size 14, class b
In the Esquisse: 3La table des sutures du groupe d’ordre 1685Courte marche à travers la théorie de Galois6Quatre groupes à double vie9Immeubles et réseaux11Le revêtement double et le miroir13Orientation et charge14Les continus15La tour assemblée18Exceptionnel veut dire relevableÉp.L’horizon : dessins d’enfants
In the volume: IIBlind ObserversVFour Reports, Six LettersVITime as CountVIIThe Branchial TreeXIIIThe Level-Seven ShadowXIVWhy OctonionsXVSpin from the Double CoverXVIThe QuartetXVIIOne Point of the Cayley PlaneXIXRulial InvariantsXXIIOne SpeedXXIVA Number Nature Could Refute