La trinité de Galois

Trois groupes, trois géométriesThree groups, three geometries

A subgroup of index $p$ in $\PSL(2,p)$ has order $(p^2-1)/2$, prime to $p$, and Dickson’s list of subgroups leaves only $A_4$, $S_4$ and $A_5$, of orders 12, 24 and 60; so $p$ is 5, 7 or 11. Each point stabilizer is the rotation group of a regular solid, and $p^2-1$ is the order of its double cover in $\mathrm{SU}(2)$: the exceptional actions are cut out by the regular polyhedra.

For $p=5$ the subgroups of index $p$ form one conjugacy class, for $p=7$ and $p=11$ two, exchanged by an element of $\PGL(2,p)$ outside $\PSL(2,p)$. One class or two is the difference the type law will read at the prime where Galois’s geometry appears.

La table des sutures d’$A_5$The seam table of the icosahedral group

In $\PSL(2,5)$ every class of subgroups is determined by the isomorphism type of its members, so the stabilizer class of an orbit can be read in any life of $A_5$ without fixing a marking, and every seam between orbits with the same stabilizer type is built by the stabilizer principle. The seam table has five columns: the five letters, $\Proj^1(\F_5)$, the icosahedron, $\Proj^1(\F_4)$ and the group itself.

The icosahedron’s column is classical. The object of size 12 is its vertices, 20 its faces, 30 its edges, 15 the golden rectangles of its edge axes, 6 its vertex axes, 10 its face axes, and the object of size 5 its five inscribed cubes, which are the five maximal sets of commuting involutions. The table extends the dictionary of chapter 6 from four objects to all nine.

Le groupe d’ordre 660The group of order 660

$\PSL(2,11)$ is the group of the 660 Möbius transformations of $\Proj^1(\F_{11})$ of determinant 1. By Galois’s theorem it has two classes of subgroups $A_5$, of eleven members each. One is fixed and called $A_5^a$; changing the choice changes the marking by an outer automorphism. The subgroups $S_3$ of a member of $A_5^a$ form one class, $S_3^a$, and those of a member of $A_5^b$ the other.

The outer automorphism, induced by $z\mapsto-z$, fuses only the two classes of elements of order 11. A pair of classes it exchanges therefore meets every class of elements equally often, since neither $A_5$ nor $S_3$ contains elements of order 11: the two objects of such a pair have one permutation character, and no seam joins them.

Le biplan comme objetsThe biplane as objects

Take the eleven members of $A_5^a$ as points and the eleven members of $A_5^b$ as blocks, with $G$ acting by conjugation, and call a point and a block incident when they meet in a subgroup of order 12. This is the biplane of Galois’s eleven-point action: five points on each block, any two points on exactly two blocks, any two blocks meeting in two points.

Pairs of points and pairs of blocks are incarnations of one rigid object, with stabilizer $D_{12}$, so a unique seam joins them: it sends two points to the two blocks through them. A triple of points on a block lies on that block only, since two blocks meet in two points, so its stabilizer lies in a member of $A_5^b$.

La table de PSL(2,11)The table of PSL(2,11)

Every object except one is incarnated on the line itself, by subsets or perfect matchings of $\Proj^1(\F_{11})$. The exception is the object of size 60. A subgroup $C_{11}$ fixes one point and cycles the other eleven, so an invariant subset is a union of those two orbits, with stabilizer $11{:}5$ or $G$, and no invariant matching exists, since the fixed point would be matched with a fixed point. The two orbits of eleven perfect matchings are Kostant’s icosahedral partitions: as a set for a subgroup $A_5$ the line is the vertex set of an icosahedron, and the antipodal pairs form a matching it fixes.

Klein supplies two more columns. The $G$-invariant cubic forms on the five-dimensional representation make a line, spanned by Klein’s cubic $x_1^2x_9+x_9^2x_4+x_4^2x_3+x_3^2x_5+x_5^2x_1$, and $X(11)$ lies on it. $X(11)\to X(1)$ is a Galois covering with group $G$, branched over three points with cyclic stabilizers of orders 2, 3 and 11, so its elliptic points incarnate the objects of sizes 330 and 220, its cusps the object of size 60, and Riemann–Hurwitz gives $2g-2=-2\cdot660+330+2\cdot220+10\cdot60=50$, genus 26. The object of the cusps is not rigid, $\Aut_G(G/C_{11})\cong C_5$, so between the cusps and Kostant’s sixty elements of order 11 there are five seams, not one.

Les pôles des solidesThe poles of the solids

Let $H$ be Galois’s subgroup of index $p$, with preimage $2T$, $2O$ or $2I$ in $\SL(2,p)$. A subgroup $H'$ of the other class, or for $p=5$ another member of the one class, has two orbits on the $p$ conjugates of $H$, of sizes 1,4; 3,4; 5,6, with stabilizers $A_4,C_3$; $D_8,S_3$; $A_4,D_{10}$. Read through the solids: the octahedral group acts on the seven points as on its three four-fold axes and four three-fold axes, the icosahedral group on the eleven as on its five inscribed cubes and six five-fold axes. For $p=7$ and $p=11$ the small orbits are the lines of a Fano plane and the blocks of the biplane.

Two more family statements are computed. $G=HZ$ with $H\cap Z=1$ for a Sylow $p$-subgroup $Z$, which is Kostant’s reading of Galois’s theorem, since for $p>11$ that subgroup has no complement. And the McKay correspondence: for each faithful character of degree 2 of $2T$, $2O$, $2I$, the graph joining $\chi_i$ to $\chi_j$ as often as $\chi_j$ occurs in $\rho\chi_i$ is the extended Dynkin diagram $\tilde E_6$, $\tilde E_7$, $\tilde E_8$, with the character degrees as its eigenvector of eigenvalue 2.

La loi sur la trinitéThe law on the trinity

Each member gets its lattice by one construction. Let $u$ be $z\mapsto z+1$, $w=(1+\sqrt{p^*})/2$, and $\chi$ the irreducible character of degree 3, 3, 5 with $\chi(u)=w$, $w-1$, $w-1$. Let $m\in\PGL(2,p)\setminus G$ be an involution and $\alpha$ conjugation by $m$, so that $\sigma\circ\chi=\chi\circ\alpha$. In the permutation module on the cosets of a subgroup $H_0$ normalized by $m$, of type $C_2$, $C_3$, $D_{10}$, the vectors $c_j=\sum_g\chi(g^{-1})e_{gj}$ span a module affording $\chi$, and their span $M$ over $\mathcal O=\Z[w]$ is a $G$-stable lattice. The map $T(x)=\sigma(P_mx)$ satisfies $T(c_j)=c_{mj}$ and $Tg=\alpha(g)T$: an explicit semilinear seam over $\alpha$ on the lattice itself.

So every statement about the residues is a statement about the reduction of $T$. For $p=7$ the lattice $M$ is Klein’s lattice. For $p=11$ it is the only $G$-stable lattice up to scaling, and by Roulleau’s computation it is the period lattice of the intermediate Jacobian of Klein’s cubic threefold, with its principal polarization; it has no vectors of norm 1 or 2, and its 110 vectors of norm 3 lie on the 55 lines fixed by the centralizers of the involutions.

Le biplan modulo 3The biplane at three

At a prime $\mathfrak P$ over 3 the residue is $\mathrm{PG}(4,3)$. Each member of one class of $A_5$ fixes exactly one point of it and no hyperplane, each member of the other exactly one hyperplane and no point, and at $\bar{\mathfrak P}$ the classes exchange roles. The point of $x$ lies on the hyperplane of $B$ exactly when $x$ and $B$ are incident in the biplane, so the biplane is drawn in $\mathrm{PG}(4,3)$ as eleven points and eleven hyperplanes, and the polarity of the law exchanges them.

The other primes give other pictures. At 5 no subgroup $A_5$ fixes a point or a hyperplane, the residue restricted to $A_5$ being the Steinberg module, and the six Borel subgroups of each $A_5$ give a frame, six points of a normal rational curve. At 2 the form reduces to a hermitian form over $\F_4$, so $G\subset\mathrm U(5,2)$, and again no $A_5$ fixes a point. At $\sqrt{-11}$ the twelve points of the rational normal quartic form the orbit with stabilizer $11{:}5$: this is $\Proj^1(\F_{11})$ itself.

Le motif de la trinitéThe pattern of the trinity

For $p=5$ the orbits of $A_5$ on the 21 points of $\mathrm{PG}(2,4)$ are $1+5+15$, with stabilizers $A_5$, $A_4$, $V_4$, and on its lines $5+6+10$, with stabilizers $A_4$, $D_{10}$, $S_3$: Galois’s five points are a conic, and the point fixed by $A_5$ is its nucleus. For $p=7$, at the prime $(w)$ over 2 the seven points of $\mathrm{PG}(2,2)$ are fixed by the members of one class of $S_4$ and the seven lines by the other; at $(w-1)$ the roles are exchanged, and $\beta$ is a polarity of the Fano plane. At $\sqrt{-7}$ the orbits are $8+21+28$, with stabilizers $7{:}3$, $D_8$, $S_3$.