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  1. Home > Articles & Issues >
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  3. Reflection factoriza ...
Conference paper

Reflection factorizations of Singer cycles

J.B. Lewis (1), V. Reiner ORCID (1), D. Stanton (1)
(1) School of Mathematics
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Imported on
November 21, 2016
Published on
December 31, 2013
Last modified on
March 31, 2025
Volume
DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)
Proceedings
DOI
10.46298/dmtcs.2401
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https://about.hal.science/hal-authorisation-v1
Reflection Group Combinatorics
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Reflection factorizations of Singer cycles

J.B. Lewis (1), V. Reiner ORCID (1), D. Stanton (1)
(1) School of Mathematics
Abstract
English
The number of shortest factorizations into reflections for a Singer cycle in $GL_n(\mathbb{F}_q)$ is shown to be $(q^n-1)^{n-1}$. Formulas counting factorizations of any length, and counting those with reflections of fixed conjugacy classes are also given.
French
Nous prouvons que le nombre de factorisations de longueur minimale d’un cycle de Singer dans $GL_n(\mathbb{F}_q)$ comme un produit de réflexions est $(q^n-1)^{n-1}$. Nous présentons aussi des formules donnant le nombre de factorisations de toutes les longueurs ainsi que des formules pour le nombre de factorisations comme produit de réflexions ayant des classes de conjugaison fixes.
Keywords
  • [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]
  • [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]
  • Singer cycle
  • Coxeter torus
  • anisotropic maximal torus
  • reflection
  • transvection
  • factorization
  • finite general linear group
  • regular element
  • q-analogue
  • higher genus
  • Coxeter element
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