forced gap

Floor 3, Ce qui est su · introduced in Chapter 3, La table des sutures du groupe d’ordre 168

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.

Plate 3.7The closed seam table, fifteen objects against seven theories: a gold disc where a basic figure realizes the class, a hatched circle where only composite figures do. The graphs have no forced gap.

Definition

The basic figures of the theories of the seam table are their simplest figures: in the Fano plane, the configurations, that is, sets of points and lines; on the projective line, its subsets; in the group, elements and subgroups under conjugation; in the Klein plane, the points of $\Proj^2(\C)$ ; in the graphs, sets of vertices of the Coxeter graph; in the octonions, the thirty lattices $L_C$ and the subalgebras spanned by units; in Thurston’s congruence link complement $M$ , its points and its cusps.

A class has a forced gap in a theory when it is realized there only by composite figures, so that the basic figures cannot reach it. The notion makes the negative space of the table a complete, checkable list.

Theorem(The forced gaps) computed

The stabilizer classes of the basic figures are exactly the following. (a) Configurations of the Fano plane: 1, $C_2$ , $V_4^a$ , $V_4^b$ , $S_3$ , $D_8$ , $S_4^a$ , $S_4^b$ , $G$ ; sets of points alone give only $S_3$ , $D_8$ , $S_4^a$ , $S_4^b$ , $G$ . (b) Subsets of $\Proj^1(\F_7)$ : $C_3$ , $C_4$ , $S_3$ , $A_4^a$ , $A_4^b$ , $7{:}3$ , $G$ . (c) Elements and subgroups of $G$ : $C_3$ , $C_4$ , $S_3$ , $C_7$ , $D_8$ , $7{:}3$ , $S_4^a$ , $S_4^b$ , $G$ . (d) Points, and lines, of $\Proj^2(\C)$ : 1, $C_2$ , $C_3$ , $C_4$ , $S_3$ , $C_7$ , $D_8$ . (e) Sets of vertices of the Coxeter graph: all fifteen classes. (f) Lattices $L_C$ and subalgebras spanned by units: $A_4^b$ , $7{:}3$ , $S_4^a$ , $S_4^b$ , $G$ . (g) Points and cusps of $M$ : 1, $C_2$ , $C_3$ , $S_3$ , $A_4^a$ , $A_4^b$ , $7{:}3$ .

Proof

(a), (b) and (e) by machine, over all $2^{14}$ configurations, all $2^8$ subsets and, for each class, unions of orbits of a representative. The reasons are short. On the projective line the subsets of sizes $0,1,\dots,8$ have stabilizers $G$ ; $7{:}3$ ; $S_3$ ; $C_3$ ; $C_4$ , $A_4^a$ or $A_4^b$ ; $C_3$ ; $S_3$ ; $7{:}3$ ; $G$ . In the Fano plane, for $H$ of class $C_3$ , $C_4$ , $A_4^a$ or $A_4^b$ every orbit of $H$ on points and on lines is an orbit of $N_G(H)$ , and for $C_7$ and $7{:}3$ the orbits are all points and all lines, so a configuration fixed by $H$ is fixed by a larger group. (c) and (d): the stabilizer of an element under conjugation is its centralizer and that of a subgroup its normalizer, and no point of $\Proj^2(\C)$ has stabilizer $V_4$ , $A_4$ , $7{:}3$ , $S_4$ or $G$ . (f) The subalgebras spanned by units are $\R$ , the seven $\R+\R e_x$ , the seven quaternion subalgebras and $\Oct$ , with stabilizers $G$ , $S_4^a$ , $S_4^b$ , $G$ , and the lattices give $G$ , $S_4^a$ , $A_4^b$ , $7{:}3$ . (g) A point of $M$ lies in one open cell, and its stabilizer acts on that cell by rotations: the point stabilizers are 1, $C_2$ , $C_3$ and $A_4$ inside tetrahedra, $C_3$ at the centres of faces, and $C_3$ and $S_3$ on edges; the cusps have class $7{:}3$ .

Remark

Every forced gap is filled by a composite figure of the same theory: a frame, an ordered triple, a pair of commuting involutions, a self-polar triangle, a face with an edge. Only the Coxeter graph reaches every class with sets of vertices. A forced gap is an empty fibre of the orbit-type description, and the marks detect it.

Examplecomputed

Representations have forced gaps too. In the faithful half $\mathbf 4$ of the Weil representation of $\SL(2,7)$ the vectors fixed by the unipotent group are the multiples of $\delta_0$ , whose stabilizer is larger; so no vector has stabilizer the odd lift of $C_7$ , and the new object of size 48 is a forced gap of the $\mathbf 4$ . Its imprint is the monomial basis of the principal series $\mathbf 8$ , whose forty-eight vectors $\zeta_6^ke_c$ are that object.

Built from: absenceincarnationstabilizer class
Objects: the object of size 42, class athe object of size 42, class bthe object of size 24the object of size 168
In the Esquisse: 10La table en deux, en sept et à l’infini11Le revêtement double et le miroir