Le revêtement double et le miroir

Les objets nouveauxThe new objects

The group of order 168 has a double cover $\tilde G=\SL(2,7)$, with $\pi\colon\tilde G\to G$ a surjection with kernel $\{\pm I\}$, and every $G$-set is a $\tilde G$-set through $\pi$. A $\tilde G$-set is pulled back if $-I$ acts on it trivially, and a transitive one that is not is a new object. Part One showed why new objects must exist: the stabilizer of a point of the seven-point action of $\SL(2,7)$ is the binary octahedral group, and the spin bundle built from it is not equivariantly trivial.

The automorphisms of a new object map onto those of its quotient, with kernel $\{1,-I\}$. On the nonzero vectors they are the scalars $\F_7^\times\cong C_6$: $-1$ is the action of $-I$, and the squares map isomorphically onto the automorphism group $C_3$ of the object of size 24, so the extension splits. On the object of size 112, $\sigma(\bar v,\bar w)=(\bar w,\overline{-v})$ generates the automorphisms and $\sigma^2$ is $-I$; below, $\sigma$ induces the exchange $(a,b)\mapsto(b,a)$ of ordered pairs of points, an involution, and every automorphism over it has order 4. So a cycle of seams whose monodromy on the ordered pairs is the exchange has monodromy $\sigma^{\pm1}$ on the object of size 112, and going round twice gives $-I$, not the identity: the sign of a spinor, met in a seam system. On the regular object the automorphisms form $\tilde G$ acting on bases from the right, and $\tilde G\to G$ does not split, $\tilde G$ having no subgroup of order 168.

On a new object the functions odd under $-I$ form a representation whose constituents are faithful: $\C[16]^-\cong4\oplus\bar4$, $\C[48]^-\cong4\oplus\bar4\oplus8\oplus8$, $\C[112]^-\cong2(4\oplus\bar4\oplus6_+\oplus6_-\oplus8)$, and $\C[336]^-$ holds each faithful irreducible representation as often as its dimension. The spin bundle over the seven points, an object pulled back from $G$, has sections $6_\pm\oplus8$; the two smallest faithful representations, 4 and $\bar4$, are carried instead by the smallest new object, the sixteen square classes.

La représentation de WeilThe Weil representation

A second theory for the new objects must be one in which $-I$ acts nontrivially, and for odd $q$ the group $\SL(2,q)$ has a classical one on the functions on $\F_q$, the Weil representation (Howe; Gérardin). For $q=7$ let $\chi$ be the Legendre symbol, $u_b=\left(\begin{smallmatrix}1&b\\0&1\end{smallmatrix}\right)$, $d_a=\left(\begin{smallmatrix}a&0\\0&a^{-1}\end{smallmatrix}\right)$, $w=\left(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right)$ and $\sqrt{-7}=\zeta+\zeta^2+\zeta^4-\zeta^3-\zeta^5-\zeta^6$, and put

So Klein’s representation is the odd half of the Weil representation, and the factor $-1/\sqrt{-7}$ in Klein’s matrix $\rho(s)$ is the one that makes $W$ a representation; it is $\rho$ itself and not its conjugate, its character at $u_1$ being $\zeta+\zeta^2+\zeta^4=(-1+\sqrt{-7})/2$. The even half is the faithful 4, and its vectors are the second theory of the new objects. The orbit of $\delta_0$ is sixteen vectors $\pm v_c$, one pair over each point $c$, with stabilizer the odd lift $\{d_au_b:a\in\{1,2,4\}\}$ of $7{:}3$: the object of size 16. The vectors fixed by the odd lift of the class $C_3$ generated by $d_2$ form the plane of $\delta_0$ and the constant function 1; off its two lines a vector has the object of size 112 as orbit, and $W(w)$ exchanges the lines with eigenvalues $\pm i$, so the automorphism group $C_4$ acts on the orbit of an eigenvector by the scalars $i^k$. The vector $\delta_0+2(\delta_1+\delta_6)$ has the regular orbit. But no vector of the 4 has stabilizer of class $C_7$, the vectors fixed by the unipotent group being the multiples of $\delta_0$: the object of size 48 is a forced gap of the 4. Its imprint is in another representation. Let $B$ be the upper triangular group, $U$ its unipotent subgroup and $\alpha$ the character of $B$ with $\alpha\bigl(\left(\begin{smallmatrix}a&b\\0&a^{-1}\end{smallmatrix}\right)\bigr)=\zeta_6^k$ for $a=3^k$. Then $\operatorname{Ind}_B^{\tilde G}\alpha$ is irreducible of dimension 8, $-I$ acts as $-1$, and the orbit of $e_\infty$ is the set of the 48 vectors $\zeta_6^ke_c$, with stabilizer $U$, the odd lift of $C_7$: a seam from the nonzero vectors of $\F_7^2$. The same construction explains all four objects over the points: for a subgroup $K$ of the torus $B/U\cong C_6$, $\tilde G/UK$ is the orbit of a basis vector of the representation induced from a character with kernel $UK$, the eight points for $K=C_6$, the 24 vectors up to sign for $C_2$, the 16 square classes for $C_3$ and the 48 vectors for $K=1$; the new ones are those with $-I\notin K$.

Seize vecteurs et une matrice de PaleySixteen vectors and a Paley matrix

With $\langle f,g\rangle=\sum_xf(x)\overline{g(x)}$, which $W$ preserves, the sixteen vectors have norm 1 and Gram matrix $I+C/\sqrt{-7}$, where $C$ is a skew conference matrix: $C_{cc}=0$, $C_{cd}=\pm1$ for $c\neq d$, $C^{\mathsf T}=-C$ and $C^2=-7I$. After changing the signs of some $v_c$ it is $\pm$ the Paley matrix, $C_{\infty x}=1$, $C_{xy}=\chi(y-x)$. And $\sqrt{-7}\,v_c=\sum_dC_{cd}v_d$, so on the real span $\R^8$ of the $v_c$, in which they are orthonormal for the real part of the form, $\tilde G$ acts by signed permutations and multiplication by $i$ is $C^{\mathsf T}/\sqrt7$.

So the smallest faithful real representation of $\tilde G$, of dimension 8, is monomial: it is $\R[16]^-$, the functions on the sixteen square classes that are odd under $-I$, and its complex structure is a Paley matrix. Read through the octonions, the eight basis lines of $\Oct$ are the eight points of $\Proj^1(\F_7)$, and the complex structure is left multiplication by the normalized sum of the imaginary units.

Le réseau $E_8$ et ses demi-racinesThe lattice E8 and its half-roots

The complex structure is integral on a lattice. For each class of tetrahedra of $M$, the vectors $\pm v_c$ and $\tfrac12(\pm v_a\pm v_b\pm v_c\pm v_d)$, over the tetrahedra $\{a,b,c,d\}$ of that class, are the 240 minimal vectors of a lattice $E_8$, namely $\Z^8+\tfrac12C_T$ for the code $C_T$ of the Steiner system, and the signed permutations preserve it. Its roots form three orbits, of sizes 16, 112 and 112: the new object of size 16 and two copies of the new object of size 112.

The complex structure tells the two orbits of half-roots apart. Let $r=\tfrac12\sum_{x\in T}\epsilon_xv_x$ be a half-root over a tetrahedron $T$. Off $T$, the vector $Cr$ is $\pm2$ at one point and 0 at the other three for the half-roots of one orbit, $O_1$, the point being the one outside $T$ fixed by the stabilizer of $r$; for the other orbit, $O_2$, it is $\pm1$ at all four points. The stabilizer of $r$ has order 3, fixes two points of the line and permutes the other six in two 3-cycles, and $T$ is a union of its orbits, so exactly one fixed point, $t(r)$, lies in $T$.

Both orbits are the object of size 112, whose automorphism group is $C_4$, its element of order 2 being the action of $-I$; so there are four seams from one orbit to the other, and the question is which of them the structure supplies. Compare the Weil vectors, where the automorphism of order 4 of an orbit of eigenvectors of $W(w)$ is multiplication by $i$. On $E_8$ multiplication by $i$ is not available, since $C/\sqrt7$ is irrational, and its integral shadow is $Q$: keep the part of $Cr=\sqrt7\,ir$ off the support, and halve it. So the two copies of the object of size 112 are joined by the reflection seam, with trivial monodromy, and by the sign seam, with monodromy $-I$, and the complex structure supplies the generator of the automorphism group.

Une famille d’algèbres sur le complexe des étoilesA family of algebras on the complex of stars

The derivations of the octonions form a Lie algebra $\mathfrak g_2$ of dimension 14; let $\mathfrak{su}(3)_x$ be its subalgebra of derivations killing $e_x$, of dimension 8. Fix one of the 42 bijections. The stabilizer in $\tilde G$ of the pair $\{\infty,0\}$ preserves $\mathfrak{su}(3)_0$, so transporting it by the signed permutations defines an algebra over each of the 28 pairs of points, and exactly seven of these consist of derivations of the octonions: those over the pairs $\{\infty,x\}$, where the algebra is $\mathfrak{su}(3)_x$. The family is equivariant over the object of size 28, so it is built. Its description as algebras of derivations of the one table exists only over the seven pairs through the point sent to 1, because only the odd lift of the stabilizer of that point acts by automorphisms: an absence with a carrier imprint, the octonionic data at seven pairs not being the restriction of an octonionic structure over all 28, and the equivariant family carrying them.

The family is itself a seam system. Call two pairs adjacent when they share a point: the Johnson graph $J(8,2)$, with 168 edges, whose 280 triangles of pairs through a common point are the 2-cells of a 2-complex $\mathcal K$, the complex of stars. Along the edge from $\{c,d\}$ to $\{c,e\}$ transport by the odd lift $u$ of the element of order 7 fixing $c$ with $d\mapsto e$. Conjugation by $u$ carries the algebra over $\{c,d\}$ onto that over $\{c,e\}$. Around every 2-cell the transports compose to the identity, so the connection is flat; in a gauge along a spanning tree every link variable lies in the stabilizer of order 12 of the base pair, and the holonomy group is all of it; and every holonomy element acts on the algebra over the base pair by an inner automorphism, commuting with $L_{e_0}$ on the complement of $\mathrm{span}(1,e_0)$, with $-I$ acting trivially, so through a group of order 6.

The star of a point is simply connected, two stars meet in exactly one vertex and no three meet, so $\pi_1(\mathcal K)$ is the fundamental group of the complete graph on the eight points, free of rank 21, and a loop at $\{a_0,a_1\}$ traces a closed walk on the points. The holonomy exchanges $a_0$ and $a_1$ exactly when the walk has odd length. Around a triangle $\{a,b\},\{b,c\},\{c,a\}$ it is the involution exchanging $a$ with $b$ and $c$ with its harmonic conjugate with respect to $a$ and $b$; around a 4-cycle it fixes $a_0$ and $a_1$, with multiplier at $a_1$ the square of the cross-ratio $(a_0,a_2;a_1,a_3)$ of its diagonals. The triangles generate the fundamental group, and each acts by a harmonic involution. The two statements are cases of one law, which holds in $\SL(2,7)$.

L’holonomie intérieure est forcéeInner holonomy is forced

The connection uses one choice of transports. To decide whether its holonomy is inner by necessity, fix the criterion first. A holonomy element $h$ fixing the base pair acts on $\mathfrak{su}(3)_0$ by an inner automorphism if it commutes with $J_0=L_{e_0}$ on the complement $V_6$ of $\mathrm{span}(1,e_0)$, and by an outer one, complex conjugation followed by an inner automorphism, if it anticommutes with $J_0$. $\GL(2,7)$ acts on the sixteen square classes, hence on $\R^8$ by signed permutations; call an element of $\PGL(2,7)$ proper if it lies in $\PSL(2,7)$, of square determinant, and improper otherwise.

So inner holonomy is forced, though not because complex conjugation is impossible. The symmetries that could conjugate the algebra over a pair, those of non-square determinant, do not preserve the family of algebras at all. The prediction registered before the computation was that improper transports would give outer holonomy around odd loops; it failed for this reason, and the alternative registered with it, inner holonomy forced, is what holds.

La table et son miroirThe table and its mirror

An improper element carries the family off itself; here is where it goes. Through a bijection $\beta$ of the sixteen vectors with the octonion basis, the octonion product pulls back to a product on $\R^8$ with unit $v_\infty$. Write $\theta=C^{\mathsf T}$ for multiplication by $\sqrt{-7}$, $\ell_c=\C v_c$, $E=\ell_\infty^\perp$, a complex space of dimension 3, and $\eta_0=\zeta+\zeta^2+\zeta^4$, $\eta_1=\zeta^3+\zeta^5+\zeta^6$ for the Gauss periods, so that $\sqrt{-7}=\eta_0-\eta_1$. An algebra and its opposite, with $x*y=yx$, have the same derivations. In the classical description of the octonions along a unit imaginary octonion, as $\C\oplus\C^3$, the product is fixed by the hermitian structure and a complex volume form on $\C^3$; the theorem says that the table and its mirror have the same hermitian structure on $\R^8$ and volume forms differing by the phase $w$, which has infinite order. The improper element is not this rotation: it anticommutes with $\theta$, so it is antilinear, and it exchanges the two tables.