The Elementary Divisors of the Incidence Matrices of Skew Lines in PG(3,p)

The elementary divisors of the incidence matrices of lines in $PG(3,p)$ are computed, where two lines are incident if and only if they are skew.


Introduction
Let V be a 4-dimensional vector space over the finite field F p of p elements, where p is a prime. We declare two 2-dimensional subspaces R and S to be incident if and only if R ∩ S = {0}. Ordering the 2-dimensional subspaces in some arbitrary but fixed manner, we can form the incidence matrix A of this relation. In this paper we compute the elementary divisors of A viewed as an integer matrix. In order to introduce some useful notation, we will view this setup as a special case of a more general situation. For brevity, an r-dimensional subspace will be called an r-subspace in what follows.
More generally, let V be an (n+1)-dimensional vector space over the finite field F q , where q = p t is a prime power. Let L r denote the set of r-subspaces of V . Thus L 1 denotes the points, L 2 denotes the lines, etc. in P(V ). An r-subspace R ∈ L r and an s-subspcace S ∈ L s are incident if and only if R ∩ S = {0}. The incidence matrix with rows indexed by the r-subspaces and columns indexed by the s-subspaces is denoted A r,s . We can and do view A r,s as the matrix of a homomorphism of free Z-modules η r,s : Z Lr → Z Ls that sends an r-subspace to the (formal) sum of all s-subspaces incident with it. Then computing the elementary divisors of A r,s is the same thing as finding a cyclic decomposition of the cokernel of η r,s . These matrices A r,s have been studied by several authors. Their pranks can be deduced from the work [5], but their elementary divisors have been computed only in the case that either r or s is equal to one [4,2].
We return now and for the remainder of the paper to the case where V is 4-dimensional over the prime field F p , and A = A 2,2 .
Theorem 1.1. The elementary divisors of the incidence matrix A are all p-powers, and are as given in the table below.
Remark. The fact that the elementary divisors are p-powers follows from more general work of Andries Brouwer [1]. Indeed, the problem that is the subject of this paper was suggested by Brouwer to the second author at the recent workshop on invariants of incidence matrices held at the Banff International Research Station.
The rest of the paper will be organized as follows. In the next section, the determinant of A will be computed. It will follow at once from this computation that the cokernel of η 2,2 is a finite p-group. We will then introduce a certain p-filtration of a GL(4, p)-module that allows for a convenient reformulation of our problem. The proof of the theorem relies on the submodule structure of this GL(4, p)-module.

The Determinant of A
Let x ∈ L 2 . Then we have as follows by an elementary counting argument. Keeping in mind that x is incident with y if and only if x ∩ y = {0}, from the above we immediately get the relation which reduces to where I and J denote the |L 2 |×|L 2 | identity matrix and all-one matrix, respectively. Since all matrices involved in (2.1) are symmetric and commute with each other, we can simultaneously diagonalize to obtain a system of |L 2 | equations: where λ 1 , λ 2 , . . . are the eigenvalues of A counted with multiplicity. Solving the quadratic equations, we see that λ 1 is either p 4 or p−p 2 −p 4 , and that for each i ≥ 2 we have that λ i is either p or −p 2 . Since the all-one vector is an eigenvector for A with eigenvalue p 4 , we must have that λ 1 = p 4 . Using the fact that the trace of A is zero, one can easily solve for the multiplicities of the remaining eigenvalues of A.
The eigenvalue p has multiplicity p 4 + p 2 and the eigenvalue −p 2 has multiplicity p 3 + p 2 + p. This yields:

Some Modules for GL(4, p)
Set G = GL(4, p). If we fix a basis of V then G acts transitively on the sets L r (r = 1, 2, 3) and the incidence map η 2,2 : Z L 2 → Z L 2 is clearly a homomorphism of ZG-permutation modules. Reduction mod p induces an F p G-permutation module homomorphism, denoted Thus we have a descending filtration We have an induced p-filtration With a little thought, one sees that for each i ≥ 0 the multiplicity of p i as an elementary divisor of A is precisely dim Fp (M i /M i+1 ). This reformulation of our problem allows us to focus our attention on the F p G-submodule structure of F L 2 p , which we now do.
For r = 1, 2, 3 define We use 1 to denote the element x∈Lr x ∈ Z Lr , and the same symbol also for its image in F Lr p (r will always be clear from the context). Since |L r | ≡ 1 (mod p) we have the decomposition Thus the essential task is to understand Y 2 . Crucial for our analysis is the F p G-submodule structure of Y 1 and Y 3 , which is well-known and given in the theorem below.
For i = 1, 2, 3 let S i denote the degree i(p − 1) component of the graded algebra S * (V )/(V p ), the quotient of the symmetric algebra on V by the ideal generated by p-powers. S i is a simple F p G-module with dimension equal to the coefficient of x i(p−1) in the expansion of (1 + x + · · · + x p−1 ) 4 . Explicitly, we have Theorem 3.1 (Cf. [4], Theorem 2 and the comment following it.).
(1) Y 1 is uniserial with composition series and simple quotients (2) Y 3 is uniserial with composition series and simple quotients The following two lemmas provide all of the information about the submodule structure of Y 2 that we will need in order to prove Theorem 1.1 in the next section. Before we can state the lemmas we must first define two important maps. Let φ : Z L 1 → Z L 2 be the ZG-module homomorphism defined by for all x ∈ L 1 .
(So φ is just η 1,2 .) The other map we call ψ : Z L 3 → Z L 2 , and it is the ZG-module homomorphism given by Denote their reductions mod p by φ and ψ, respectively. Notice how these maps respect the decomposition (3.1). In particular, and these are all submodules of Y 2 . This observation will be used frequently.
Let L be a simple, non-trivial F p G-module. By Frobenius reciprocity we have where P is the stabilizer of a 2-subspace. Thus if L is a simple submodule of Y 2 , then by dualizing we see that the multiplicity of L in soc(Y 2 ) is dim Fp (L * ) P . Therefore P fixes a nonzero vector of L * , and it follows from the general theory of modular representations of finite groups of Lie type [3,6] that L * is determined up to isomorphism by this property. Since P has a fixed point on S 2 , we thus have S 2 ∼ = L * . Furthermore the fixed vector is unique up to scalars, so soc(Y 2 ) is simple. Since S 2 and Y 2 are both self-dual, (1) follows.
Finally, since e 4 = 1 it follows easily that f 4 = 1 and f i = 0 for i ≥ 5.