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In this section, we will consider the actions of graph and field automorphisms on some of theS2∗-candidate subgroups. Some of these computations have been done already:
(i) The graph and field automorphisms of thep-restricted representations of SL2(q) are considered in Corollary 4.3.7.
(ii) The graph and field automorphisms of the spin representations are considered in Lemma 4.3.27, Lemma 4.3.28 and Lemma 4.3.29.
We also do not have results on the action of the field automorphism on rewrit- ten tensor product groups preserving an orthogonal form, so we will not include these here.
We start the section with some general results, before providing specific com- putations for the remainingS∗
2-candidate subgroups. The following will be useful throughout.
Lemma 4.4.1. [8, Lemma 5.1.6] There is a natural embedding ofXˆl(q) intoXˆl(qs)
for any s≥1, and the restriction of any p-restricted module forXˆl(qs) to Xˆl(q) is
Proof. Note that both ˆXl(q) and ˆXl(qs) arise from restrictions of ˆXl(Fp) to cen-
tralisers of the field automorphismsφe and φes respectively; in particular this gives us the natural embedding of ˆXl(q) into ˆXl(qs) sinceφes∈ hφei. This proves the first
part. If we have ap-restricted module of ˆXl(Fp) with weight λ, then its restriction
to ˆXl(q) and ˆXl(qs) will be p-restricted modules of the respective groups, also with
weightλ, hence the second part holds.
Corollary 4.4.2. Let Ge = ˆXl(pe) be a group of Lie type over the field Fpe, and
supposeGe has an absolutely irreducible representation whose image is contained in
an untwisted classical group Ωe over Fpe, for every e≥1. Then the automorphism
φG of Ge is induced by the field automorphism φΩ of Ωe.
Proof. First note that Ωe has a subgroup Ω1 obtained by restricting the field of definition, and this in turn has a subgroup G1. Note also that we can obtain a subgroup ofGeby restricting the field in the natural representation of Ωe, and from
Lemma 4.4.1 it follows that this subgroup is alsoG1.
The field automorphismφG of Ge is induced by φΩ followed by conjugation by a matrix g overFpe; denote this automorphism by cg. Since G1 is fixed by φG, it follows thatφΩcg centralisesG1.
We also haveφΩ centralises Ω1, and henceφΩ also centralisesG1. Thus since the representation of G1 is absolutely irreducible, it follows thatg =λIn for some
scalarλ∈F∗pe, so thatφG is induced byφΩ.
We first consider those groups which are tensor powers of the natural module of SLn(qd). For these, the following general results will be useful:
Lemma 4.4.3.LetG= SLn(qd), letV be the naturalG-module with basise1, . . . , en,
and let ρ : G → Ω := SLnd(q) be a representation with corresponding module
W := V ⊗Vσ ⊗. . .⊗Vσd−1 and image Gρ := ˆG which can be rewritten over
Fq by conjugation by a matrix c∈GLnd(qd). Then:
(i) For g ∈ G, the matrix M(g) of the action of g on the rewritten tensor field representation is given by c−1(g⊗gσ⊗. . .⊗gσd−1)c.
(ii) For n > 2 and q odd, the automorphism γG is induced by γΩ followed by
conjugation by cTc, and in terms of outer automorphisms of Ω is induced by
γΩ if n≡0,1 mod 4 or if n≡2 mod 4and d >2 and by γΩδΩ otherwise.
(iii) For q odd, the automorphism φG is induced by φΩ followed by conjugation by
c−φc, and in terms of outer automorphisms of Ω is induced by φΩδ
(p−1,nd) (2,n)
Ω if
Proof.
(i) This is clear from the construction of the module. (ii) We have that
M(g)γΩ= (c−1(g⊗gσ⊗. . .⊗gσd−1)c)−T =cT(g−T ⊗g−σT ⊗. . .⊗g−σd−1T)c−T
whereas
M(gγG) =c−1(g−T ⊗g−σT ⊗. . .⊗g−σd−1T)c,
and so M(gγG) = c−1c−TM(g)γΩcTc. Thus the automorphism γ
G of G is in-
duced by the product of the automorphism γΩ and the automorphism induced by conjugation bycTc.
To show thatcTc∈GLnd(q), we have thatcTcinduces a module isomorphism betweenγGW andWγΩ, both of which are absolutely irreducible modules over
Fq; hence by Lemma 1.7.12 (i) there must exist λ∈Fqd such that λc−1c−T ∈ GLnd(q). We then have λc−1c−T = (λc−1c−T)σ = λqc−σc−σT. Rearranging, we get that Ind =λq−1cc−σc−σTcT =λq−1cc−σ(cc−σ)T.
By Lemma 2.3.2 we can rescalecsuch thatcc−σis a permutation matrix; hence (cc−σ)T = (cc−σ)−1 and soλq−1 = 1. Thus λ∈
Fq and so c−1c−T ∈GLnd(q); hence so is its inversecTc.
In order to decide which automorphism conjugation by cTc induces, we need
to find det(cTc). We have that cc−σ is the permutation matrix from Lemma 2.2.1, which we denotegσ, and the conditions onnanddgiven in the statement
determine whether detgσ is 1 or−1. If detgσ = 1, this means that det(c)q−1 =
1 and so det(c)∈Fq. In particular we have detcTc= detc2 is a square inFq.
Hence by Lemma 3.2.14 γG is induced by a conjugate of γΩ. If detgσ = −1
then we have detcq−1 = −1, so that detc /∈
Fq (since elements of Fq have
multiplicative order dividing q−1) but detc2 ∈ Fq (since det(c2)q−1 = 1); in particular this means that det(cTc) is not a square in Fq, and so again by
Lemma 3.2.14 we have that γG is induced by a conjugate ofγΩδΩ. (iii) Takingφas thep-power automorphism, we have that
whilst
M(gφ) =c−1(gφ⊗gφσ⊗. . .⊗gφσd−1)c,
and soM(gφ) =c−1cφM(g)φc−φc. Thus we have that the automorphismφGof
Gis induced by the automorphism φΩ, followed by conjugation by the matrix
c−φc.
To show that c−φc is written over Fq, recall from the proof of part (ii) that
cc−σ = gσ is a permutation matrix. In particular cc−σ ∈ GLnd(p), so that
cc−σ = (cc−σ)φ = cφc−σφ. Rearranging, we get that c−φc = c−σφc−σ = (c−φc)σ, so thatc−φc∈GLnd(q), and so as beforec−φcinduces some power of the diagonal automorphism. Note that here the conditions of Lemma 3.2.14 will not generally hold, so we must compute directly.
We have det(c−φc) = det(c)1−p. When detgσ =−1, we have that detc /∈Fq,
but detc2 ∈Fq, so that detcp−1∈Fqis a (p−(21,n,n)d)-power of a primitive element in Fq, and hence φG is induced by φΩδ
(p−1,nd) (2,n)
Ω . When detgσ = 1 it follows
as before that detc ∈ Fq, and so when q is odd the determinant of c−φc is a
(p−1, nd)-power. Recall that Out(Ω) =hδΩ, γΩ, φΩ|δ(q −1,nd) Ω =γ 2 Ω =φeΩ= [γΩ, φΩ] = 1, δγΩΩ =δΩ−1, δΩφΩ =δpΩi. It follows that a typical element of Out(Ω) can be written in the formγΩiφjΩδΩk, andφγ i Ωφ j Ωδ k Ω Ω =φ δk Ω
Ω . The last relation in the automorphism group gives us that
φ−Ω1δΩφΩ =δpΩ, so thatδΩφΩδΩ−1 =φΩδΩp−1 and so φ δk Ω Ω =φΩδk(1 −p) Ω . Thus the conjugates of φΩ are of the form φΩδi(p
−1)
Ω and thus consist of the product of the automorphismφΩ by matrices whose determinants are a (p−1,(q−1, nd))- power, and sincep−1|q−1, this is the same as matrices whose determinants are a (p−1, nd)-power. Thus from above, when detgσ = 1 we have that φG
is induced by a conjugate of φΩ. We saw above that when detgσ =−1, φG is
induced by φΩδ
(p−1,nd) (2,n)
Ω , and it remains to see when this is conjugate to φΩ. When n ≡ 3 mod 4 then (p(2−1,n,n)d) = (p−1, nd) so in this case detc−φc is a (p−1, nd)-power, so that φG is induced by a conjugate ofφΩ as well.
It remains to prove whether φΩ and φΩδ
(p−1,nd) (2,n)
Ω are conjugate when n ≡ 2 mod 4 and d = 2, so we are considering these as elements of Out(SLnd(q)).
Similarly to before we have that (φΩδ (p−1,nd) (2,n) Ω )γ i Ωφ j Ωδ k Ω =φΩδ pj(−1)i(p−1,nd) (2,n) +(1−p)k Ω ,
and this is conjugate to φΩ if and only if we can find i, j, k, m∈Z such that pj(−1)i(p(2−1,n,n)d) =m(pe−1, n2)−k(1−p). Ifp≡3 mod 4 then the left hand side is odd and the right hand side is even for every choice of i, j, k, m, whilst ifp ≡1 mod 4 then the right hand side is divisible by 4 whilst the left hand side is divisible by 2 but not 4. Hence φΩ and φΩδ
(p−1,nd) (2,n)
Ω are not conjugate.
The below result will be sufficient for our purposes when the representation ofG preserves a unitary form.
Lemma 4.4.4.LetG= SLn(q2), letV be the naturalG-module with basise1, . . . , en,
and let ρ : G → SLn2(q2) be a representation with corresponding module W :=
V⊗(V∗)σ, whereσ denotes theq-power field automorphism ofG. LetM(x) denote the matrix of the action of x∈G onW. Then:
(i) M(x) =x⊗x−σT, and M(x∗) =M(x)∗.
(ii) Gρ < Ω := SUn2(q, gσ), where gσ is the permutation matrix from Corollary
2.2.2.
(iii) The automorphism φG is induced by φΩδΩ ifn≡2 mod 4 andp≡3 mod 4,
and φΩ otherwise.
Proof.
(i) The first part is clear from the definition of W. For the second, note that
M(x)∗ = (x⊗x−σT)∗=x−σT ⊗x=M(x∗).
(ii) Since (W∗)σ = W, it follows from Lemma 1.7.8 that Gρ must preserve a unitary form. We have
M(x)−σT =M(x)∗=x−σT ⊗x=gσ−1(x⊗x−σT)gσ =g−σ1M(x)gσ,
where gσ is the permutation matrix from Lemma 2.2.1 which acts on W by
permuting the tensor factors. Rearranging gives us thatgσ =M(x)gσM(x)σT,
so that M(x) preserves the unitary form gσ.
(iii) In the notation of Lemma 3.2.16, we may setα=φG andβ =φΩ. Then since
Fp, we have that L = In2 and λ= 1 (using the notation of Lemma 3.2.16). Hence, by Lemma 3.2.16 we have κ = 1 and the automorphism inducing φG
depends on whether detg
1−p
2
σ is equal to 1 or −1. When n ≡ 0,1,3 mod 4,
detgσ = 1 and so in this caseφG is induced by a conjugate ofφΩ. Whenn≡2 mod 4, we have detgσ =−1 (since in the notation of Lemma 2.2.1,d= 2), so
that when p≡1 mod 4 φG is induced by a conjugate ofφΩ, and when p≡3 mod 4 φG is induced by a conjugate ofφΩδΩ.
Lemma 4.4.5.LetGbe anS2∗-candidate subgroup ofΩ = SL16(3e)with composition
factorL4(3e) as in Proposition 4.3.14. Then the automorphisms γΩ andφΩ induce
γG andφG respectively on G.
Proof. The fact that φΩ induces φG follows from Corollary 4.4.2. We use a similar
construction to consider the action of the graph automorphism.
Note that Ω has a subgroup ˆΩ isomorphic to SL16(3) by restricting the field of definition, and this in turn has an S2-subgroup ˆG = SL4(3). Note also that ˆG is a subgroup ofGobtained by restricting the field in the natural representation of SL4(3e). The duality automorphism γG of G is induced by γΩ followed by conju- gation by some matrixg overF3e. The action of γΩcg on Grestricts to the action of the duality automorphism γGˆ on ˆG, since duality on G restricts to duality on
ˆ
G. Similarly, the automorphism γGˆ of ˆGis induced by γΩˆcˆg for some matrix ˆg over
F3. Thus γΩcg and γΩˆcˆg both induce γGˆ on ˆG, and since γΩ induces γΩˆ on ˆΩ it follows that g and ˆg must be the same up to scalar multiplication. Since scalar multiplication does not affect the conjugation action of the matrix, we may assume thatg= ˆg and it suffices to consider the action in the case whereq= 3. Computer computations insl4gammashow that det ˆg = 1 so thatγΩˆ induces γGˆ on ˆG and so
γΩ induces γG on G.
Lemma 4.4.6. Let G be an S∗
2-candidate subgroup of Ω = SU16(3e) with composi-
tion factorU4(3e)as in Proposition 4.3.15. Then the automorphismφΩ inducesφG
onG.
Proof. Note that G and Ω occur naturally as subgroups of ¯G= SL4(32e) and ¯Ω = SL16(32e) respectively; furthermore, since by Proposition 4.3.15 the form preserved by Ω andGis over the base field F3, the field automorphismsφG¯ andφΩ¯ restrict to the field automorphismsφG and φΩ respectively. From Lemma 4.4.5 we have that
Proposition 4.4.7. Let G be an S2∗-candidate subgroup of Ω = SL16(q) with com-
position factorL4(q2)as in Proposition 4.3.17. Then the automorphismsγΩ andφΩ
induceγG and φG respectively on G.
Proof. This is direct from the table of candidates and Lemma 4.4.3.
Proposition 4.4.8. Let G be an S∗
2-candidate subgroup of Ω = SU16(q) with com-
position factorL4(q2) as in Proposition 4.3.18. Then the automorphismφΩ induces
φG onG.
Proof. This is direct from the table of candidates and Lemma 4.4.4.
Lemma 4.4.9. LetGbe anS∗
2-candidate subgroup ofΩ = Sp16(q) with composition
factorS4(q)as in Proposition 4.3.21. Then the automorphismφΩ inducesφG onG.
Proof. Direct from Corollary 4.4.2.
4.5
Containments
In this section we check containments of S2∗-candidate subgroups in other S∗
2-candidate subgroups. Note that we only have one S2∗-candidate subgroup in dimension 17, and no candidates as subgroups of Ω−16(q), so we have no containments to check in these cases.
We begin with some useful general results.
Lemma 4.5.1. [48, Lemma 10.3.1] Let G=Z(G).S be a quasisimple group, where
S is a nonabelian simple group, and suppose G < H. Then for some K < Z(G),
K.S embeds into a nonabelian composition factor of H.
Lemma 4.5.2. Let Ωbe a classical group, and GandH groups with corresponding faithful representations ρG and ρH respectively, such that GρG and HρH are sub-
groups ofΩ. Suppose we have a containmentHρH ≤GρG, and thatρH is irreducible
(respectively absolutely irreducible). Then for any groupM withH ≤M ≤G, there exists an irreducible (respectively absolutely irreducible) representationρM ofM with
M ρM <Ω.
Proof. ρM = ρG|M, and clearly ρM is irreducible since ρM|H =ρH, an irreducible
representation. IfρH is absolutely irreducible, thenρM must also be absolutely irre-
ducible; otherwise, by Lemma 1.7.3 there exist non-scalar matrices which centralise
Corollary 4.5.3. Let G and H be quasisimple groups, with corresponding faith- ful absolutely irreducible representations ρG and ρH respectively such that HρH <
GρG <Ω for some classical groupΩ. Suppose additionally that there is a contain-
ment ofH inside a subgroup C of G. Then C must only be centralised by scalars.
Proof. By Lemma 4.5.2, there must exist an absolutely irreducible faithful repre- sentationρC ofC which embeds into Ω. SinceρC is absolutely irreducible it follows
from Lemma 1.7.3 that CρC is centralised by scalars in Ω, and thus from the con-
struction ofρC we must have that C is also centralised by scalars.
Proposition 4.5.4. Let Ω = Sp16(q). Then there are no containments between
S2∗-candidate subgroups of Ω.
Proof. From Theorem 4.2.1 we see that the S2∗-candidate subgroups are SL2(q) for
p≥17 and Sp4(q) forp6= 2,5, so the only possibility for containment is whenp≥17 and SL2(q)<Sp4(q).
From [8, Section 8.2, Table 8.12] we see that Sp4(q) has a number of sub- groups isomorphic to SL2(q). By Corollary 4.5.3 for there to be a containment the subgroup must only be centralised by scalars. Hence for instance there is no containment of SL2(q) inside the C2-subgroup Sp2(q)2 : 2 since any copy of SL2(q) would be centralised by Sp2(q). The other C2-subgroup of Sp4(q), GL2(q).2, has an index-2 subgroup which is reducible in the natural representation of Sp4(q) and hence is centralised by non-scalar elements; thus the 16-dimensional representation of GL2(q) will also not be only centralised by scalars, ruling out a containment here also. This, along with Lagrange, rules out all possible containments except the S-subgroup of Sp4(q).
From Theorem 4.3.3 and Proposition 4.3.6, this subgroup is the action group ofS3(W) whereW is the natural module of SL2(q), andS3(W) has highest weight 3. We obtain the 16-dimensional representation of Sp4(q) by taking a submodule of the tensor productV4⊗V5whereV4 is the natural module of Sp4(q) andV5 is the natural module of Ω◦5(q), these groups being isomorphic. Again by Theorem 4.3.3 and Proposition 4.3.6, SL2(q) embeds absolutely irreducibly into Ω◦5(q) by the module
S4(W) with highest weight 4; hence by Lemma 4.1.21 the SL2(q)-module obtained as the restriction of the 20-dimensional moduleV4⊗V5 has highest weight at most 7, and thus we have the same restriction on the highest weight of the SL2(q)-module obtained as a restriction of the 16-dimensional submodule ofV4⊗V5. The module corresponding to the 16-dimensionalS2∗-candidate subgroup SL2(q) of Ω isS15(W) with highest weight 15, and hence these are not the same representation. Hence
there is no containment of SL2(q) inside Sp4(q) when considered as S2∗-candidate subgroups of Ω.
Proposition 4.5.5. Let Ω = Ω+16(q). Then we have the following containments:
(i) L2(q2).2<S4(q2).2for all q, and the normaliser ofL2(q2).2is not maximal in
any almost simple extension of Ω contained in CGO+16(q).
(ii) S4(q2).2<Sp8(q) when q is even, and the normaliser of S4(q2).2 is not maxi-