[Paper Review] The Elementary Divisors of the Incidence Matrix of Skew Lines in PG(3,q)

Andries E. Brouwer, Joshua E. Ducey+1 more|arXiv (Cornell University)|Mar 1, 2011

graph theory and CDMA systems4 citations

TL;DR

This paper computes the Smith normal form of the incidence matrix A of skew lines in PG(3,q), where two lines are incident if they are skew. It shows that all elementary divisors are powers of p, and provides explicit formulas for their multiplicities using combinatorial data over F_q, with key results derived from character sums and p-adic valuation techniques applied to the matrix's structure over finite fields.

ABSTRACT

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

Motivation & Objective

• To determine the elementary divisors of the incidence matrix A of skew lines in PG(3,q), where incidence is defined by trivial intersection.
• To compute the Smith normal form of A as an integer matrix, focusing on its p-adic structure.
• To derive explicit formulas for the multiplicities of elementary divisors p^i in terms of combinatorial data over F_q.
• To extend partial results on elementary divisors of incidence matrices A_{r,s} to the case r=s=2 in PG(3,q).
• To establish a connection between the matrix's p-adic invariants and character sums over tuples in [3]^t with specified counts of entries.

Proposed method

• The incidence matrix A is interpreted as the adjacency matrix of the non-collinearity graph on the Klein quadric, leveraging its strongly regular graph structure with known parameters.
• The matrix A satisfies a quadratic matrix equation involving I, J, and A, which allows computation of its eigenvalues and thus the p-adic valuation of det(A), implying all elementary divisors are powers of p.
• Elementary divisor multiplicities e_i(A) are computed via a formula involving tuples in [3]^t with exactly i entries equal to 2, weighted by coefficients d(𝐬) derived from the expansion of (1+x+...+x^{p-1})^4.
• The multiplicities e_{2t+i} for 0 ≤ i ≤ t are given by e_{2t+i} = ∑_{𝐬∈ℋ(i)} d(𝐬), where d(𝐬) is a product of coefficients from the p-ary expansion of the polynomial (1+x+...+x^{p-1})^4.
• The proof relies on representation theory and module decompositions over group algebras, using characters and submodules to relate the matrix structure to the p-adic elementary divisors.
• A key step involves showing that A_{r,1}A_{1,s} ≡ −A_{r,s} (mod p^t), which links the structure of the product matrix to the original incidence matrix and enables the use of character sum techniques.

Experimental results

Research questions

• RQ1 What are the elementary divisors of the incidence matrix A of skew lines in PG(3,q), where incidence corresponds to trivial intersection of 2-dimensional subspaces?
• RQ2 How can the multiplicities of the elementary divisors p^i be computed explicitly in terms of q = p^t?
• RQ3 What is the role of the p-adic valuation in determining the structure of the Smith normal form of A?
• RQ4 How do combinatorial structures such as tuples in [3]^t with specified numbers of 2s relate to the invariant factors of A?
• RQ5 Can the general theory of incidence matrices A_{r,s} be extended to cases where r=s=2 in PG(3,q), especially in terms of p-adic invariants?

Key findings

• The elementary divisors of A are all powers of p, as shown by the eigenvalue structure of A and the fact that |det(A)| is a power of p.
• The multiplicities e_i of p^i satisfy symmetry: e_i = e_{3t−i} for 0 ≤ i < t.
• For 0 ≤ i ≤ t, the multiplicity e_{2t+i} is given by ∑_{𝐬∈ℋ(i)} d(𝐬), where ℋ(i) is the set of tuples in [3]^t with exactly i entries equal to 2.
• In the case p=3, t=2 (i.e., q=9), the multiplicities are e_4=202, e_5=256, e_6=361, and e_8=1, with e_0=361, e_1=256, e_2=6025, and e_3=202 by symmetry.
• When p=2, the sum in Theorem 2.2 simplifies significantly due to d(𝐬)=0 for tuples containing adjacent 1 and 3 or 1 at start and 3 at end.
• The formula e_i(A_{r,s}) = ∑_{𝐬∈Γ(i)} d(𝐬) for 0 ≤ i < t generalizes the result to arbitrary r,s, with Γ(i) defined via character sum conditions.