Absolute order in general linear groups

TL;DR

This paper explores the absolute order in general linear groups, revealing its properties and providing explicit formulas for certain intervals, with implications for understanding group structure and combinatorics.

Contribution

It introduces two equivalent descriptions of the absolute order on GL(V) and analyzes its properties, including self-duality and explicit enumeration formulas over finite fields.

Findings

Absolute order has two equivalent characterizations.

Intervals from identity to regular elliptic elements have simple enumeration formulas.

The order exhibits self-duality and other structural properties.

Abstract

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.