Remark.
Observe that the group P\Omega6-(q) is treated as
PSU4(q) and
P\Omega5(q) as PSp4(q).
The group PSp4(2) is not simple, when one encounters it one should
take its derived group. Observe that this group can be ignored
too since PSp4(2)' is isomorphic
to PSL2(9) and
Alt6, whence has been studied well in the past.
It seemed unnatural to exclude it tough.
The groups P\Omega3(q), P\Omega4-(q)
and P\Omega6+(q) will not be considered
as they are isomorphic to linear groups, and
the groups
P\Omega2+(q),
P\Omega2-(q) and
P\Omega4+(q) are not simple.
We try to determine the multiplicity free permutation
representations and the distance-transitive graphs.
The notation in those tables is as follows:
For the multiplicity-free column:
a + means it is multiplicity free,
a ? means it is not know
(and for the moment don't try to answer the question)
no entry means that one should try to decide it since we
are often dealing with an explicit group which is relatively small,
(i.e., either a + or we should remove the entry).
Information
can be obtained from the person mentioned in the column labeled person.
Theorem Let S be a simple classical group, not the linear group. Let G be acting on the polar associated polar space of S with soc(G) = S and let H be a maximal subgroup of G not containing S. If the permutation character is multiplicity free, then H is the stabilizer of a totally singular subspace or S and the type of H are as in Table C.1 or H is of type C9 and S and F*(H) are as in Table C.2.
| Table C.1 | ||||||
| S | type H | class | mpf | comments | person | dtg |
| PSUn(q) | stabilizer of non isotropic point | C1 | + | Jan/John | solved | |
| PSU6(q) | SU3(q) x SU3(q) | C2 | ? | open: q<17 | ||
| PSU4(q) | SU2 (q) x SU2 (q) | C2 | ? | q not 2 | van Bon | open: q=2, 3 |
| PSU4(2) | stabilizer of a basis | C2 | + | rank 3 | ATLAS | yes |
| PSU5(2) | stabilizer of a basis | C2 | + | GAP: rank 7 | Breuer | ? |
| PSU3(3) | stabilizer of a basis | C2 | + | GH(2,2) | ATLAS | yes |
| PSU3(4) | stabilizer of a basis | C2 | + | ATLAS: rank 5 in Aut. | ? | |
| PSUn(q) | Spn(q) | C5 | + | n even | Jan/John | solved |
| PSU4(q) | O4-(q) | C5 | ? | q odd | ? | |
| PSU4(3) | normalizer of Z421+4 | C6 | + | ATLAS: rank 5 | ? | |
| PSU3(5) | normalizer of 32:Q8 | C6 | + | GAP, rank 12 | Breuer | ? |
| PSp2n(q) | stabilizer of a hyperbolic line | C1 | + | n>2 | Jan/John | solved |
| PSp2n(q) | Spn(q) x Spn(q) | C2 | ? | n even, see also remark 3 | van Bon/Inglis | open: q even and n= 4,..,12 |
| PSp2n(q) | Spn(q2) | C3 | + | n even | John | ? |
| PSp2n(q) | Un(q) | C3 | ? | q odd | John | open: n=3 |
| PSp12(q) | Sp4(q3) | C3 | q=2,3,4 | ? | ||
| PSp6(2) | Sp2(23) | C3 | + | atlas, rank 6 | ? | |
| PSp6(q) | Sp2(q3) | C3 | q=4,3,5,8 | ? | ||
| PSp4(3) | normalizer of 2-1+4 | C6 | + | atlas, rank 3 | - | yes |
| PSp4(p) | normalizer of 2-1+4 | C6 | p= 5,7 | ? | ||
| PSp8(3) | normalizer of 2-1+6 | C6 | ? | |||
| PSp2n(q) | O2n+(q) | C8 | + | q even | Inglis | solved |
| PSp2n(q) | O2n-(q) | C8 | + | q even | Inglis | solved |
| P\Omega2n+(q) | non isotropic point | C1 | + | Jan/John | solved | |
| P\Omega2n+(q) | stabilizer of a O2-(q) space | C1 | + | ? | John | solved |
| P\Omega8+(q) | O4-(q) x O4-(q) | C2 | ? | under triality O4+(q2) | ? | |
| P\Omega8+(2) | O2-(2) x O2-(2) x O2-(2) x O2-(2) | C2 | + | GAP, rank 12 | Breuer | ? |
| P\Omega2n+(q) | On+(q2) | C3 | ? | n even | Jan/Nick | ? |
| P\Omega2n+(q) | On(q2) | C3 | ? | nq odd | Jan/Nick | ? |
| P\Omega2n+(q) | Un(q) | C3 | + | n even | John | open: n=6,8,10 |
| P\Omega2n-(q) | non isotropic point | C1 | + | Jan/John | solved | |
| P\Omega2n-(q) | stabilizer of a O2-(q) space | C1 | + | John | solved | |
| P\Omega2n-(q) | O-n(q2) | C3 | ? | n even | Jan/Nick | ? |
| P\Omega2n-(q) | On(q2) | C3 | ? | nq odd | Jan/Nick | ? |
| P\Omega2n-(q) | Un(q) | C3 | + | n odd | John | open: n=5 |
| P\Omegan(q) | non isotropic point | C1 | + | nq odd | Jan/John | solved |
| P\Omegan(q) | stabilizer of a O2-(q) space | C1 | + | nq odd | John | solved |
| P\Omega7(3) | stabilizer of a basis | C2 | + | GAP, rank 11 in Aut. | Breuer | ? |
| P\Omega9(3) | stabilizer of a basis | C2 | ? | |||
| Table C.2 | |||||
| S | F*(H) | mpf | comments | person | dtg |
| PSU3(3) | PSL2(7) | + | rank 3 in aut | - | yes |
| PSU3(5) | Alt7 | + | rank 3 HoSi | - | yes |
| PSU3(5) | Alt6 | + | lines HoSi | - | yes |
| PSU3(5) | PSL2(7) | + | GAP rank 9 | ? | |
| PSU4(3) | PSL3(4) | + | GAP, rank 3 | - | yes |
| PSU4(3) | Alt7 | + | GAP, rank 6 | ? | |
| PSU4(5) | Alt7 | ? | |||
| PSU4(5) | PSU4(2) | ? | |||
| PSU6(2) | PSU4(3) | + | rank 3 | - | yes |
| PSU6(2) | M22 | + | GAP rank 8 | ? | |
| PSU9(2) | J3 | ? | |||
| PSp4(3) | Alt6 | + | rank 3, see also remark 2 | - | yes |
| PSp4(q) | Sz(q) | + | 2 < q an odd power of 2 | Jan/John | solved |
| PSp4(7) | Alt7 | ? | |||
| PSp6(q) | G2(q) | + | q even | Jan/John | solved |
| PSp6(5) | J2 | ? | |||
| PSp8(2) | Alt10 | + | rank 5 | Inglis | ? |
| PSp12(2) | Alt14 | Nick notes not selfpaired | no | ||
| PSp12(3) | Suz | ? | |||
| P\Omega7(q) | G2(q) | + | Jan/John | solved | |
| P\Omega7(3) | PSp6(2) | + | rank 4 | ? | |
| P\Omega7(p) | PSp6(2) | p=5,7 | ? | ||
| P\Omega7(3) | Alt9 | + | GAP rank 6 | ? | |
| P\Omega9(3) | Alt10 | ? | |||
| P\Omega8+(q) | P\Omega7(q) | + | q odd, under triality non-isotropic point | - | solved |
| P\Omega8+(q) | PSp6(q) | + | q even, under triality non-isotropic point | - | solved |
| P\Omega8+(2) | Alt9 | + | rank 4 | ? | |
| P\Omega8+ (3) | P\Omega8 +(2) | + | rank 4 | ? | |
| P\Omega8+(p) | P\Omega8+(2) | p= 5,7 | ? | ||
| P\Omega10+(3) | Alt12 | ? | |||
| P\Omega14+(2) | Alt16 | Nick notes not selfpaired | no | ||
| P\Omega10-(2) | Alt12 | + | rank 6 in Aut grp | ? | |
| P\Omega12-(2) | Alt13 | Nick notes not selfpaired | no | ||
Remark 1 Observe that for n=8 the action of P\Omega8+(q) on Un(q) becomes under triality a stabilizer of a O2- -space.
Remark 2 In the table of C9 groups the case S = PSp4(3) = PSU4(2) and F*(H) = Alt6 is listed only once since Alt6 as a subgroup of PSU4(2) is Sp4(2)' and is of type C5. The case S = PSp4(2) and F*(H) = Alt5 is not listed since Alt5 = P\Omega4-(2) and is thus of type C8.
Remark 3
We justify the + entries in tables and make some
observations about the open cases.
The cases where H is a maximal parabolic have been studied in the
past.
Suppose S= PSUn(q).
If H a stabilizer of a non isotropic point
the permutation character is multiplicity free.
If H of type PSpn(q) it follows from
the thesis of Nick Inglis.
If H of type PSU2(q) x PSU2(q), then this is
the action of P\Omega6-(q) on
O2-(q)-spaces.
This action is multiplicity free if the group is large enough.
The remaining ones follow from the ATLAS or are well known.
Suppose S = PSp2n(q).
If H the stabilizer of a hyperbolic line, then if
q=2 it follows from the known literature.
If H of type PSpn(q2),
P\Omega{2n} ^\epsilon (q)
it follows from
the thesis of N. Inglis that
the permutation charter is multiplicity free.
If H of type (Spn(q) x Spn(q)), n even,
the techniques as developed
in the thesis of N. Inglis can be used to show that for 2n=8 and 12
(2n=4 is well known) all orbits are self paired and for 2n \geq 16
there exists a non self paired
orbit. Whence the permutation charter is multiplicity free for
2n=4,8,12, but still undecided in general.
Finally suppose that S is isomorphic to an orthogonal group.
The stabilizers of non isotropic
points have been studied by M.W. Liebeck, C.E. Praeger and J.Saxl
If H of type PSUn(q), then it follows from
the thesis of N. Inglis
that the permutation charter is multiplicity free.
If H is the stabilizer of an O2--space the multiplicity
freeness will be proved.
It is also known that if S = P\Omega6+(q) and
H of type O3(q), with q odd, then the permutation charter
is not multiplicity free as follows from work on the linear groups.
We also have to study the permutation characters and distance-transitive graphs that arise from maximal subgroups that are normalized by the graph automorphism for PSp4(q)', q even, and the triality automorphism for P\Omega8+(q). The results on permutation characters is summarized in the next 2 theorems.
Theorem Let G be an almost simple group with socle P\Omega8+(q) Suppose H is a maximal subgroup of G normalized by a triality automorphism and H \not \leq P\Delta8 +(q). If the permutation character is multiplicity free, then H \cap P \Omega8+(q) is as in Table C.3.
| Table C.3 | ||||||
| group no. | H \cap P\Omega8+(q) | mpf | comments | person | dtg | |
| 5 | stabilizer of a totaly singular line | + | Rank 5 in Aut(P\Omega8+(q)) | Jan/John | solved | |
| 15 | G2(q) | ? | might be +, many orbs same length, c.f. Jan | |||
| 26 | ^( 1/d (q+1) x 1/dGU3(q)).2d | |||||
| 56 | ^( 1/d Zq+1)4.d3.23.S4 | q=2,3,4 | ||||
| 72 | P\Omega8+(2) | + | q=3, rank 4 | |||
| 72 | P\Omega8+(2) | q=5,7 | ||||
| 62 | P\Omega8+(2) | q=23 | ||||
Remark The two + entries are both well known. R. Lawther (personal communication) showed that if G=P\Omega8+(q) and H = G2(q), then the permutation character is multiplicity free only for q=2,3 and that in the other cases there is a character occuring with multiplicity 2. What the behaviour is under automorphisms is not known.
Theorem Let S= PSp4(q)' with q even. Let G be almost simple with socle S and contain the graph automorphism. Let H be a maximal subgroup of G, normalized by and containing the graph automorphism. The permutation character is multiplicity free if and only if H is the stabilizer of a flag, q > 2 is an odd power of 2 and H \cap S = Sz(q), q > 2 is an even power of 2 and H \cap S = PSp4({\sqrt q}), or q=2 and H \cap S= 5:2 or H \cap S = 32:4. In all cases the DTG problem is solved.
Remark
If H is of type Sz(q) or
of type PSp4(\sqrt q) the multiplicity-freeness
follows from work by R. Lawther.