Group actions and geometric combinatorics in 𝔽qd

A.1 Quadratic forms, spheres and isotropic subspaces

Let k be a field of characteristic not equal to 2. Let V be a finite-dimensional k-vector space and Q a quadratic form on V. Associated to Q, there is a symmetric bilinear inner product 〈⋅,⋅〉:V×V→k such that Q is the norm map Q⁢(x)=〈x,x〉. We say that Q is non-degenerate if 〈x,y〉=0 for all y∈V implies x=0. The isometry group of Q is denoted O⁢(Q), the orthogonal group associated to Q, and consists of all linear transformations which preserve (the inner product associated to) Q. When Q is the standard dot product on 𝔽qd, then the isometry group is denoted Od⁢(𝔽q).

The quadratic form Q can be represented by a symmetric matrix 𝔹 such that Q⁢(x)=xT⁢𝔹⁢x and under change of basis 𝔸, 𝔹 transforms as 𝔹→𝔸T⁢𝔹⁢𝔸 and so the determinant of 𝔹 modulo squares is an invariant of Q called the discriminant, denoted by disc⁡(Q). If Q is non-degenerate, we view the discriminant as an element of k*/(k*)2.

When k is a finite field of odd characteristic, Witt showed (see for example [9] or [12]) that two quadratic forms Q,Q′ are isomorphic if and only if they are equal in dimension (via the vector space they are defined on) and discriminant. Thus on 𝔽qd there are only two distinct non-degenerate quadratic forms up to isomorphism: the standard dot product and another one. We will denote their isometry groups as Od⁢(𝔽q) and Od′⁢(𝔽q) for the remainder of this section.

Furthermore, if x and y are two nonzero elements of 𝔽qd, Witt also showed that there is an isometry taking x to y if and only if x and y have the same norm, i.e., Q⁢(x)=Q⁢(y). Thus for any non-degenerate quadratic form, the sphere of radius r in (𝔽qd,Q) can be identified as the quotient of O⁢(Q) by the stabilizer of any given element on that sphere. Thus a discussion of sphere sizes is equivalent to a discussion of isometry group and stabilizer sizes.

Let H denote the hyperbolic plane, the 2-dimensional quadratic form represented by matrix [0110] of discriminant -1. The hyperbolic plane H has the important property that it has two totally isotropic lines (lines consisting completely of vectors of norm 0) and its n-fold orthogonal direct sum nH has maximal totally isotropic subspaces of dimension n called Lagrangian subspaces. Furthermore, H plays an important role in the classification of non-degenerate quadratic forms over finite fields F of odd characteristic. One can characterize completely the structure of any such non-degenerate quadratic form as follows: If dim⁡(V)=2⁢n is even, then the isomorphism class of Q has two possibilities:

Here NK/F is the 2-dimensional quadratic form given by the norm map of the unique degree 2 extension field K of F. If dim⁡(V)=2⁢n+1 is odd, then

where c=(-1)n⁢disc⁡(Q).

As the standard dot product on 𝔽qd has dimension d and discriminant 1 it follows that when d is even, it is isomorphic to d2⁢H if (-1)d2 is a square modulo q and d-22⁢H⊕NK/F if not. When d is odd, it is isomorphic to d-12⁢H⊕(-1)d-12⁢x2. Regardless, when d≥3, totally isotropic subspaces are unavoidable.

The following theorem was proved by Minkowski at the age of 17 (see [10]) based on the classification we just discussed:

The degenerate case possibilities in Theorem A.1 affect sizes of spheres and stabilizers but not the magnitude of the isometry groups. This is because all non-degenerate quadratic forms in 1 dimension are nonzero multiples of each other and hence have exactly the same isometry group. Then using induction in dimension 2 and higher, and that nonzero radius sphere counts agree up to (1+o⁢(1)) in all cases from Theorem A.1, one can prove that for any two non-degenerate d-dimensional quadratic forms Q and Q′ one has |O⁢(Q)|=|O⁢(Q′)|⁢(1+o⁢(1)) as q→∞. In fact, one has:

A.2 Degenerate simplices

Let us work inside 𝔽qd, where q is odd in this section.

Let [x0,…,xk] be a k-simplex in 𝔽qd. After a translation this simplex transforms to the 0-pinned k-simplex [0,x1-x0,…,xk-x0] which has the same congruence class and ordered distance vector. It is easy to see that the stabilizer in the Euclidean group of [x0,…,xk] has exactly the same size as the stabilizer in Od⁢(𝔽q) of the pinned simplex [0,x1-x0,…,xk-x0] so we will restrict out attention to O⁢(Q) and pinned simplices in this subsection.

The dimension of a k-simplex is the dimension of the space spanned by the vertices in its corresponding pinned simplex. A k-simplex is called non-degenerate if this is as big as possible, i.e., k-dimensional, and degenerate otherwise. All k-simplices are degenerate when k>d.

Take a k-simplex and let V denote the vector space generated by its associated pinned simplex. Suppose we have a non-degenerate quadratic form Q on 𝔽qd like for example the standard dot product. Let V⊥ denote the perp with respect to Q. We then say that V is good if V∩V⊥=0. Under these conditions, Q≅Q|V⊕Q|V⊥ and it is easy to show that the stabilizer of the k-simplex in O⁢(Q) can be identified with O⁢(Q|V⊥) which has the size |Od-m⁢(𝔽q)|⁢(1+o⁢(1)), where m is the dimension of V. Thus we have proven the following lemma:

Note that non-degenerate d-simplices in 𝔽qd are automatically good as V⊥=0.

In the case the simplex is not good, in the sense that the associated spanned space has isotropic vectors, the argument of Lemma A.3 can break down, as Gram–Schmidt breaks down when one has vectors of norm 0. This situation will not lead to any big difficulty in this paper, though, so we will ignore it for now. However as an example to show that things are different, consider the line (t,i⁢t) in 𝔽q2, where q=1⁢ mod ⁢4 and the pinned 1-simplex [(0,0),(1,i)]. Under O2⁢(𝔽q), its orbit has size 2⁢(q-1) which is twice as big as orbits of non-degenerate good pinned 1-simplices. Thus the corresponding circle is twice as big as normal and so the stabilizer set is half as big as normal. Thus (1+o⁢(1)) no longer suffices to record the difference. For many arguments a factor of 2 is no big deal but in arguments involving cancelation of lead order terms, they can play a big role.

As a final goal of this section, let us show that the set of ordered distances generated by degenerate k-simplices in 𝔽qd are insignificant in the set of all ordered distances of k-simplices.