Absolute order in general linear groups

This paper studies a partial order on the general linear group GL(V) called the absolute order, derived from viewing GL(V) as a group generated by reflections, that is, elements whose fixed space has codimension one. The absolute order on GL(V) is shown to have two equivalent descriptions, one via additivity of length for factorizations into reflections, the other via additivity of fixed space codimensions. Other general properties of the order are derived, including self-duality of its intervals. Working over a finite field F_q, it is shown via a complex character computation that the poset interval from the identity to a Singer cycle (or any regular elliptic element) in GL_n(F_q) has a strikingly simple formula for the number of chains passing through a prescribed set of ranks.
Comment: 26 pages. v2: Minor edits; Question 6.3 and some references added