Number of subspaces of a Galois geometry

The Galois geometry $P G (𝑛, 𝑞)$ contains $(𝑛 - 1 𝑑 - 1) 𝑞$ subspaces of dimension $𝑑$ , #m/thm/geo/fin where

(𝑛 𝑑) 𝑞 = 𝑑 - 1 \prod 𝑖 = 0 𝑞 𝑛 - 𝑖 - 1 𝑞 𝑖 + 1 - 1

is the Gaußian binomial coëfficient.¹

Proof

We define

Θ 𝑞 (𝑟) = 𝑞 𝑟 + 1 - 1 𝑞 - 1 [𝑟, 𝑠] 𝑞 = \prod (𝑞 𝑖 - 1)

Let $𝑉 = G F (𝑞) 𝑛 + 1$ , so $P G (𝑛, 𝑞) = P (𝑉)$ . As a vector space $𝑉$ contains $𝑞 𝑛 + 1$ points. Now the number of points in $P G (𝑛, 𝑞)$ equals the number of 1-dimensional subspaces of $𝑉$ , and different subspaces have only the origin in common. Thus the number of 1-dimensional subspaces is

𝑞 𝑛 + 1 - 1 𝑛 - 1 = Θ 𝑞 (𝑛)

Each $𝑑$ -dimensional subspace is determined by $𝑑 + 1$ linearly independent points, so the number of $𝑑$ -dimensional subspaces in an $𝑛$ -dimensional projective space is the total number of independent sets of points of size $𝑑 + 1$ divided by the number of independent sets of points of size $𝑑 + 1$ in each $𝑑$ -dimensional subspace. This gives

[𝑛 - 𝑑 + 1, 𝑛 + 1] 𝑞 [1, 𝑑 + 1] 𝑞

as required.

#state/tidy | #lang/en | #SemBr

2020. Finite geometries, ¶4.7, p. 79 ↩

Number of subspaces of a Galois geometry

Footnotes