Finally, suppose that we have found all potentialS1-maximal subgroups in
Ω.R, where Ω P tSL˘npqq,Spnpqq,Ωnpqqu and R ď Outp¯Ωq. The final step is to show whether any of these S1-subgroups are contained in any other
S1-subgroup preserving the same form, because in this case they cannot be
maximal. The following is based on [8, Section 4.8, p.211].
Lemma 4.9.1. Let H1ρ1 and H2ρ2 be two S1-subgroups of Ω preserving
the same form. Assume thatH1 ďH2 and assume that there does not exist
an element gPGLnpquq such that pH1ρ2qg is defined over a proper subfield
of Fqu. If ρ2 reduces to an absolutely irreducible representation of H18 such
that ρ2 and ρ1 are equivalent on H18 then H1ρ1 cannot be maximal as an
S1-subgroup.
Proof. By definition,H1 andH2are almost simple extensions of quasisimple
groups. Therefore, H1 “H18.R1 and H2 “ H28.R2, where R1 and R2 are
subgroups of the outer automorphism groups of H8
1 and H28 respectively
such thatH1ρ1“NΩpH18ρ1q and H2ρ“NΩpH28ρ2q. By assumptionH1 ď
H2 and thereforeρ2 is also a representation of H1.
We want to show that ρ2 is an absolutely irreducible representation of
H1. By assumptionH18ρ2does not stabilise any non-zero subspace ofpFqrqn for any r. From this it follows that pH8
any subspace either. Sinceρ1 is equivalent toρ2 onH18 it follows thatH1ρ1
is notS1-maximal.
For many groups it is straightforward to show that H8
1 ρ is irreducible.
There are two easy methods which work in most cases.
Lemma 4.9.2. Let ρ be a faithful absolutely irreducible representation of H8
2 of dimension n and suppose that H18ďH28.
(i) If H8
1 has no non-trivial absolutely irreducible representation of di-
mension smaller than n then H8
1 ρ is absolutely irreducible since it cannot
be split into smaller parts. (ii) IfH8
1 has non-trivial absolutely irreducible representationsρ1, . . . , ρk
of dimensions ni, i P t1, . . . , ku, smaller than n with ř
ni “ n but there exists some g PH8
1 such that
ř
Tracepgρiq ‰Tracepgρq for every such set of representationsρ1, . . . , ρk, then H18ρ is irreducible.
5
Maximal
S
1-Subgroups in Dimension 13
In this chapter we are going to determine the S1-maximal subgroupsG in
dimension 13. We will follow the procedure described in Chapter 4. Here ΩP tSL˘13pqq,Ω˝
13pqqu. Furthermore, we will denote the conformal group of
Ω byC.
5.1 S1-Subgroups in Dimension 13
LetGbe a quasisimple group with an absolutely irreducible representation ρof dimension 13 in cross characteristic. All such groups are listed in Table 5.1.1 on p.62. We are interested in finding the extensions by automorphisms of Gρ which might be S1-maximal in some classical group. The table also
contains some useful information about these groups which we will need later.
In the first column of the table we can find the name of the group fol- lowed by its order and Schur indicator. Column ‘#ρ’ gives the number of weakly equivalent representations of ρ that do not lie in the same equiva- lence class (see Definition 4.3.2). The outer automorphisms that stabilise these representations are given in the ‘Stab’ column. The characteristics over which these representations occur can then be found in the column ‘Charc’. Here 0 stands for all prime numbers that do not divide the order of G. We also require the character rings of the representations (see Definition 2.2.2) which is given in the column ‘ChR’. In the final column we state the size of the outer automorphism group of G. The list of groups G and the characteristics of the representations where taken from [18], whereas the in- formation in the other columns is mostly from [12, 24]. The ordinary and Brauer character tables of A14 and A15 and the Brauer character tables of
S6p3q are not contained in [12, 24] and soGAP was used to determine these
character tables.
Comments on the character ring column in Table 5.1.1
(i) The algebraic irrationalities of the 13-dimensional absolutely irre- ducible representations of U3p4q are b5 and z5, but b5 “z5`z45 and hence
the character ring isZrz5s.
(ii) The respective rows of the character table of S6p3q contain algebraic
conjugates of b27, z3 and i3. However, all of them are elements of Zrz3s
since i3 “z3´z32 and b27 “ 12p´1`
?
´27q “ 12p´1`i3q `i3 “b3`i3 “
(iii) The algebraic irrationalities of the 13-dimensional absolutely irre- ducible representations of S4p5q are, apart from algebraic conjugates of b5,
algebraic conjugates of r5, but r5 “1`2b5.
Further information regarding the irrationalities can be found in Table 2.2.1 (p.19).
Table 5.1.1: Potential S1-maximal subgroups in dimension 13 Gp Order Ind #ρ Stab Charc ChR |Out|
L2p27q 22¨33¨7¨13 ˝ 2 3 0,2,7,13p‰3q Zrb27s 6 S6p3q 29¨39¨5¨7¨13 ˝ 2 1 0,2,5,7,13p‰3q Zrz3s 2 U3p4q 26¨3¨52¨13 ˝ 4 1 0,3,13p‰2,5q Zrz5s 4 A7 23¨32¨5¨7 + 1 2 3,5 Z 2 A8 26¨32¨5¨7 + 1 2 3,5 Z 2 A14 210¨35¨52¨72 + 1 2 0,3,5,11,13p‰2,7q Z 2 ¨11¨13 A15 210¨36¨53¨72 + 1 2 3,5 Z 2 ¨11¨13 L2p13q 22¨3¨7¨13 + 1 2 0,3p‰2,7,13q Z 2 L2p25q 23¨3¨52¨13 + 2 22 0,3,13p‰2,5q Z 22 L3p3q 24¨33¨13 + 1 2 0,13p‰2,3q Z 2 S4p5q 26¨32¨54¨13 + 2 1 0,3,13p‰2,5q Zrb5s 2 J2 27¨33¨52¨7 + 2 1 3 Zrb5s 2
Theorem 5.1.1. Let G be an S1-subgroup ofΩP tSL˘13pqq,Ω˝
13pqqu. Then
Gis contained in Table 5.1.1. Proof. See the tables in [18].