Symplectic groups $Sp_{2m}(2)$ as $2$-transitive permutation (i.e. Galois) groups

Question

I am looking for information about the symplectic groups $Sp_{2m}(2)$ as permutation group acting on quadratic forms.

Consider the block matrices

$$e=\begin{pmatrix}0&1\\0&0\end{pmatrix}, \qquad f=\begin{pmatrix}0&1\\-1&0\end{pmatrix}=e-e^T$$

on the vector space $(\mathbb{F}_2)^{2d}$ equipped with the standard basis. Consider the symmetric bilinear form $\phi(u,v)=ufv^T$, and let $\Omega$ be the set of all quadratic forms $\theta(u)$ such that

$$\phi(u,v)=\theta(u+v)-\theta(u)-\theta(v).$$

In particular the quadratic form $\theta_0(u)=ueu^T$ is $\in\Omega$, and any other element of $\Omega$ can be shown to be of the form

$$\theta_a = \theta_0(u)+\phi(u,a).$$

Now $Sp_{2m}(2)$ acts on $\Omega$, and it turns out that the action splits in two distinct orbits

$$\Omega^+=\{\theta_a|\theta_0(a)=0\},\qquad \Omega^-=\{\theta_a|\theta_0(a)=1\},$$

of size respectively $2^{m-1}(2^m+1)$ and $2^{m-1}(2^m-1)$. The group $Sp_{2m}(2)$ acts $2$-transitively on each of these orbits, see Chap. 7 of Permutation Groups (Dixon, Mortimer) for more details.

Question: what can be said of the action of these two sets? Here are two more specific questions: is the stabilizer of one point acting imprimitevely, for some block structure? What are the orbits of a $2$-point stabilizer?

Motivation: I am studing the Galois groups of polynomials (trinomials) over function fields in characteristic $p$, which can be proven to be $2$-transitive. This was done by Abhyankar, Galois theory on the line in nonzero characteristic (1996), which computed the Galois group of many trinomials, and I think that his results can be extended to cover more cases. If I'm wrong I will happen to have learned something about $2$-transitive groups.

The $2$-transitive permutations groups are classified (affine groups, alternating/symmetric, projective, symplectic $Sp_{2m}(2)$, unitary $PGU_3(q^3)$, Suzuki $Sz(q)$ and Ree $R(q)$, plus a few sporadic groups). Computing the local Galois group at a ramified place it is possible to describe the action of a subgroup, the inertia subgroup, as a permutation group on the roots. This allows to rule out certain familes of $2$-transitive groups, and sometimes it is possible to determine completely the Galois group. And the symplectic group at the moment is the family that I find more difficult to understand.

Answer 1

Both of these actions are 2-primitive, so the 1-point stabilizer acts primitively on the remaining points.

The 1-point stabilizers in the two actions are the orthogonal groups ${\rm SO}^{\pm}_{2m}(q)$, and the 2-point stabilizers are the maximal parabolic subgroups of these orthogonal groups with structure $2^{2m-2}.{\rm SO}^{\pm}_{2m-2}(q)$. They are the stabilizers of isotropic points in the actions of the orthogonal groups on their natural modules.

The 2-point stabilizers have 4 orbits, two of length 1. The other two are the sets of points that are perpendicular to or not perpendicular to the fixed isotropic point. They apparently have lengths $2^{2m-2}$ and $2^{2m-2} \pm 2^{m-1}$. (That last claim is based on computations with $m \le 6$, but I expect I could work it out if obliged to!)

Maurizio: To answer your query, I did all of the calculations by hand, except for the lengths of the orbits of the 2-point stabilizers, but since they are just the numbers of isotropic vectors that are or are not orthogonal to a given isotropic vector, this should be a routine calculation.

I will try and find a reference for information about these 2-transitive actions of ${\rm Sp}_{2m}(2)$. The easiest way to construct them on a computer is as the action on the cosets of their orthogonal maximal subgroups. This was straightforward in Magma, because the standard functions constructing these groups resulted in the orthogonal groups already being subgroups of the symplectic groups. Irritatingly, in GAP, the standard functions produced groups with different invariant bilinear forms, although one could be conjugated to the other by a permutation matrix.

Here is a GAP calculation to construct the 136-degree 2-transitive representation of ${\rm Sp}_8(2)$.

gap> G:=Sp(8,2);;

gap> H:=SpecialOrthogonalGroup(1,8,2);;

gap> Display(InvariantBilinearForm(G).matrix);

. . . . . . . 1

. . . . . . 1 .

. . . . . 1 . .

. . . . 1 . . .

. . . 1 . . . .

. . 1 . . . . .

. 1 . . . . . .

1 . . . . . . .

gap> Display(InvariantBilinearForm(H).matrix);

. 1 . . . . . .

1 . . . . . . .

. . . 1 . . . .

. . 1 . . . . .

. . . . . 1 . .

. . . . 1 . . .

. . . . . . . 1

. . . . . . 1 .

gap> P:=PermutationMat((2,8)(4,6),8,GF(2));;

gap> HP:=H^P;;

gap> IsSubgroup(G,HP);

true

gap> I := Image(FactorCosetAction(G,HP));;

gap> LargestMovedPoint(I);

136

gap> Orbits(Stabilizer(I,1));

[ [ 2, 17, 40, 25, 45, 21, 68, 36, 41, 14, 29, 75, 96, 27, 24, 130, 102, 70, 66, 73, 88, 23, 49, 43, 38, 11, 20, 56, 92, 42, 13, 31, 30, 133, 93, 136, 72, 8, 100, 131, 115, 129, 123, 9, 47, 52, 86, 28, 19, 89, 116, 51, 74, 105, 67, 22, 34, 6, 76, 54, 81, 7, 64, 44, 37, 3, 26, 135, 85, 108, 132, 78, 58, 61, 134, 94, 107, 113, 125, 84, 118, 104, 127, 5, 33, 98, 50, 32, 46, 114, 90, 77, 10, 48, 122, 69, 35, 18, 65, 57, 12, 99, 117, 83, 4, 55, 106, 60, 82, 59, 128, 87, 126, 109, 95, 62, 79, 111, 121, 53, 120, 80, 97, 101, 91, 71, 16, 112, 39, 15, 103, 119, 63, 124, 110 ] ]

gap> Orbits(Stabilizer(I,[1,2],OnTuples));

[ [ 3, 17, 54, 22, 50, 15, 21, 7, 42, 56, 51, 12, 18, 62, 52, 25, 23, 8, 11, 71, 44, 53, 46, 69, 49, 36, 20, 16, 68, 65, 58, 70, 38, 29, 24, 9, 19, 4, 55, 35, 13, 28, 63, 41, 48, 72, 67, 61, 33, 40, 5, 14, 26, 64, 37, 39, 60, 59, 34, 31, 10, 27, 6, 45, 43, 66, 32, 30, 47, 57 ], [ 73, 98, 117, 77, 122, 93, 101, 104, 133, 75, 110, 119, 81, 88, 78, 103, 127, 126, 107, 90, 89, 113, 102, 95, 99, 114, 123, 94, 136, 132, 131, 79, 135, 80, 108, 112, 92, 76, 83, 84, 86, 87, 106, 97, 134, 85, 116, 100, 124, 118, 111, 115, 105, 121, 96, 128, 91, 120, 129, 130, 125, 74, 82, 109 ] ]