Absolute order in general linear groups

Jia Huang, Joel Brewster Lewis, Victor Reiner

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

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 languageEnglish (US)
Pages (from-to)223-247
Number of pages25
JournalJournal of the London Mathematical Society
Volume95
Issue number1
DOIs
StatePublished - Feb 2017

Bibliographical note

Funding Information:
This work was partially supported by NSF grants DMS‐100193, DMS‐1148634, and DMS‐1401792.

Publisher Copyright:
© 2017 London Mathematical Society

Keywords

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

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