Exceptionnel veut dire relevable

Quand la loi est aveugleWhen the law is blind

A lift of a life’s module would be a complex representation of the module’s dimension, of the group or of a cover, whose character the Galois group moves. So the failures can be checked against the complex characters: either there is no representation of that dimension at all, or the ones there are cannot be moved.

For $\PSL(2,8)$ and $\mathrm U_3(3)$ no complex irreducible has the module’s dimension, and their Schur multipliers are trivial, so no cover helps; for $A_8$ neither the group nor its double cover has a representation of dimension 4. For $G_2(2)'$ and $\mathrm{Sp}_4(2)'=A_6$ lifts exist, but they are rational: the one character of degree 6 of $G_2(2)'$ is invariant under every automorphism, and the two faithful characters of degree 4 of $2{\cdot}A_6$ are exchanged by the outer automorphisms the law would have to realize.

La portée de la loiThe reach of the law

The exceptions have a common cause, and in small dimension it decides almost everything. A lattice with residue $\bar V$ gives a projective representation of $S$ in $\PGL(n,\C)$. It is faithful, because an element acting by a scalar acts by a scalar on the residue, and irreducible, because the residue is. A system of imprimitivity with $k>1$ blocks would give a homomorphism $S\to S_k$ with $k\le n\le4$; $S_k$ is solvable and $S$ is simple and non-abelian, so the map is trivial and the representation reducible. The image is therefore primitive, and the primitive finite subgroups of $\PGL(n,\C)$ for $n\le4$ are on Klein’s list for $n=2$ and Blichfeldt’s for $n=3,4$.

Les exceptions expliquéesThe exceptions explained

So a life is blind to the law in one of two ways. Either its group is too large to act projectively in its module’s dimension in characteristic zero: $\PSL(2,8)$, of order 504, in dimension 2; $\mathrm U_3(3)$, of order 6048, in dimension 3; $A_8$, of order 20160, in dimension 4. Or its lifts are rational, so the outer automorphism exchanges characters that no Galois element moves: $A_6$ as $\mathrm{Sp}(4,2)'$, and $\mathrm U_3(3)$ as $G_2(2)'$.

The lives the law sees are the remaining kind, lifts on the lists whose characters the Galois group moves, and each comes with its lattice: the icosians, Klein’s lattice, Valentiner’s lattice and $E_8$ over $\Z[\omega]$.

La droite sur $\F_q$The line over a finite field

The projective line is the first test. Its natural module has dimension 2 and, for $q$ odd, its conic module dimension 3, so a residue would make $\PSL(2,q)$ isomorphic to $A_5$, $A_6$ or $\PSL(2,7)$, and the orders decide. The values left are those of the exceptional isomorphisms, and at each of them the lattice exists: the icosians for $q=4,5$, Klein’s lattice for $q=7$, Valentiner’s for $q=9$.

Résidus, résidus quaternioniques, cœursResidues, quaternionic residues, hearts

To remove the bound on the dimension the drafts widen both sides. A life is now an isomorphism of $S$ with $X/Z(X)$ for $X$ any classical or exceptional group on its natural module: $\SL(n,q)$, the conic module of $\PSL(2,q)$, $\mathrm{SU}(n,q)$, $\mathrm{Sp}(2m,q)$, the orthogonal groups, and the exceptional groups on their minimal modules, from $G_2(q)$ in dimension 7 (or 6 in characteristic 2) to $E_8(q)$ in dimension 248. And a lift may be of three kinds: a residue as before; a quaternionic residue, the reduction of a lattice over a maximal order of a quaternion algebra at a prime where the algebra ramifies; or a heart, a reduction whose composition factors are the natural module once and the trivial module $t$ times.

Each kind gives a degree bound. Write $d_0(S)$ for the least degree of a nontrivial irreducible projective complex representation of $S$. A residue needs $n\ge d_0(S)$, a quaternionic residue a character of degree exactly $2n$, and a heart has $t\le\dim H^1(S,\bar V^*)\le n/2$, by the bound of Guralnick and Hoffman, so that $n+t\ge d_0(S)$. The least degree grows exponentially in the rank (Landazuri, Seitz and Zalesskii) while $n$ grows linearly, so only finitely many lives can lift. Kneser’s family shows the senses side by side: a root lattice modulo a prime is a residue where its form stays nondegenerate and a heart with one trivial factor where it acquires a radical.

Les vies qui se relèventThe lives that lift

The candidates are found by comparing the degree bounds with $n$ over every family, using Tiep’s lower bounds for the degrees of cross-characteristic representations and, for the groups with lives in two characteristics, the exact degrees in the tables of Hiss and Malle. No life of dimension above 26 survives. Each survivor is then realized or excluded: the residues by lattices, the root lattices among them by Kneser’s method, the Coxeter–Todd lattice $K_{12}$ modulo $\sqrt{-3}$ by direct computation, and the Tits group by Tiep and Zalesskii’s observation that its two characters of degree 26 stay irreducible modulo 2.

The lattices are familiar ones: root lattices, the Witting lattice, $K_{12}$, Klein’s and Valentiner’s lattices, the icosians, and, for one quaternionic row, the Leech lattice as a lattice over the Hurwitz order. Three reductions are close to the list but are not lives, because their images are proper subgroups: $J_2<G_2(4)$, $A_7<\GL(4,2)$ and $\mathrm{Co}_1<\Omega^+(24,2)$.

Deux caractéristiques, un réseauTwo characteristics, one lattice

The exceptional isomorphisms across characteristics are the six groups of the theorem, and the list of lifts sorts them. For $A_5$, $\PSL(2,7)$ and $\mathrm U_4(2)$ one lattice has both lives as residues, at two primes: the double life is two reductions of one object in characteristic zero. For $A_6$ and $\mathrm U_3(3)$ the same holds once quaternionic residues are admitted, through $E_8$ and $K_{12}$ over the maximal order of the quaternion algebra ramified at 3 and $\infty$. The coincidence $A_8\cong\PSL(4,2)$ is not of this kind on the four-dimensional side: that module lifts in no sense, and $A_8$ reaches it only through the Klein correspondence $\Lambda^2$, as the heart $\Omega^+(6,2)$.

$\PSL(2,8)\cong{}^2G_2(3)'$ is the one exception. Its life at 3 lifts, as the seven-dimensional Ree lattice, but its life at 2 lifts in no sense: the least degree of a nontrivial projective representation, 7, is too large for a module of dimension 2, and the Ree lattice modulo 2 has it only as a composition factor.

Ce qui reste à bâtirWhat remains to be built

The type law sees a row exactly when its lift is moved by Galois. Of the residues, rows with Galois-moved lifts include the new ones of the Witting lattice, $K_{12}$ and the Tits group; rows with rational lifts fixed by the outer automorphisms in question realize them linearly; and $\mathrm{Sp}(4,2)'$ keeps its rational lifts exchanged, which no Galois element realizes. At $\Omega(5,3)$ complex conjugation is ramified at $\sqrt{-3}$ and swaps $5a$ and $5b$, so the outer automorphism of $\mathrm U_4(2)$ is realized by a linear map, $\SO(5,3)$ acting through $W(E_6)$; the row of $K_{12}$ is of the same ramified type. For the Tits group the same holds at $\sqrt{-2}$ if its outer automorphism exchanges $26a$ and $26b$, which the ATLAS listings suggest and nothing in the drafts decides.

The next computations the drafts list: the two quaternionic rows of characteristic 2 computed rather than cited, the Leech lattice over the Hurwitz order modulo $1+i$ for $G_2(4)$ and the rank-5 lattice for $\mathrm U_5(2)$; the Tits lattice over $\Z[\sqrt{-2}]$ and the action of its outer automorphism; the type law on the new rows; and a seam reading of the exception $\PSL(2,8)\cong{}^2G_2(3)'$, residue against composition factor. Each reduction on the list is classical, but the complete list in its three senses was not found in the sources searched.