Absolute order in general linear groups

This paper studies a partial order on the general linear group (Formula presented.) called the absolute order, derived from viewing (Formula presented.) as a group generated by reflections, that is, elements whose fixed space has codimension one. The absolute order on (Formula presented.) is shown to have two equivalent descriptions: one via additivity of length for factorizations into reflections and 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 (Formula presented.), 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 (Formula presented.) has a strikingly simple formula for the number of chains passing through a prescribed set of ranks.