## Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 223-247 |

Number of pages | 25 |

Journal | Journal of the London Mathematical Society |

Volume | 95 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2017 |

## Keywords

- 05E10
- 20C33 (secondary)
- 20G40 (primary)

## ASJC Scopus subject areas

- Mathematics(all)