Ulrich Faigle asked whether this has an immediate combinatorial interpretation in the following sense: Is it always possible to partition the lattice of subspaces of GF (q) into two intervals of length n− 1 and (qn−1 − 1) intervals of length n− 2, for n ≥ 2? We show that such an interval decomposition of the lattice of subspaces of a finite-dimensional vector space exists if and only if there e...

Some connections between the rational curve K := {(1, t, t q+1 | t ∈ GF (q 2))}∪ {(0, 0, 1)} and regular symplectic spreads of P G(3, q), q even, are presented by using some geometry and combinatorics of symplectic polarities commuting with unitary polarities. Some results on the geometry of commuting orthogonal and unitary polarities will be also discussed.

Let M * (q) be the unique nonassociative finite simple Moufang loop constructed over GF (q). We prove that Aut(M * (2)) is the Chevalley group G 2 (2), by extending multiplicative automorphism of M * (2) into linear automorphisms of the unique split octonion algebra over GF (2). Many of our auxiliary results apply in the general case. In the course of the proof we show that every element of a s...

For an integer n ≥ 3, a rank-n matroid is called an n-spike if it consists of n three-point lines through a common point such that, for all k ∈ {1, 2, . . . , n− 1}, the union of every set of k of these lines has rank k + 1. Spikes are very special and important in matroid theory. Wu [13] found the exact numbers of n-spikes over fields with 2, 3, 4, 5, 7 elements, and the asymptotic values for ...