Counting Configurations of Vectors in a Finite Vector Space with an Orthogonal, Symplectic or Unitary Geometry

Abstract

Given a finite-dimensional vector space \(V\) over a finite field \(F\) of odd characteristic, and equipping \(V\) with an orthogonal (symplectic, unitary) geometry, the following two questions are considered:

1. Given some linearly independent vectors \(w_1, w_2, \ldots, w_k \in V\) and the \(k \times k\) matrix \(A = (\langle w_i, w_j\rangle)\), and given scalars \(\alpha_1, \alpha_2, \ldots, \alpha_k, \beta \in F\), how many vectors \(v \in V\), not in the linear span of \(w_1, w_2, \ldots, w_k\), satisfy \(\langle w_i, v\rangle = \alpha_i\) (\(i = 1, 2, \ldots, k\)) and \(\langle v, v\rangle = \beta\)?
2. Given a \(k \times k\) matrix \(A = (\lambda_{ij})\) with entries from \(F\), how many \(k\)-tuples \((v_1, v_2, \ldots, v_k)\) of linearly independent vectors from \(V\) satisfy \(\langle v_i, v_j\rangle = \lambda_{ij}\) (\(i, j= 1, 2, \ldots k\))?

An exact answer to the first question is derived. Here there are two cases to consider, depending on whether or not the column vector \((\alpha_i)\) is in the column space of \(A\). This result can then be applied iteratively to address the second question.