Reflection factorizations of Singer cycles

J.B. Lewis; V. Reiner; D. Stanton

doi:10.46298/dmtcs.2401

J.B. Lewis ; V. Reiner ; D. Stanton - Reflection factorizations of Singer cycles

dmtcs:2401 - Discrete Mathematics & Theoretical Computer Science, January 1, 2014, DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) - https://doi.org/10.46298/dmtcs.2401

Reflection factorizations of Singer cyclesConference paper

Authors: J.B. Lewis ¹; V. Reiner ¹; D. Stanton ¹

1 School of Mathematics

[en]
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.

[fr]
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.

https://doi.org/10.46298/dmtcs.2401

Source: HAL:hal-01207611v1

Volume: DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)

Section: Proceedings

Published on: January 1, 2014

Imported on: November 21, 2016

Keywords: [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], [en] Singer cycle, Coxeter torus, anisotropic maximal torus, reflection, transvection, factorization, finite general linear group, regular element, q-analogue, higher genus, Coxeter element

Funding:

Source : OpenAIRE Graph

Reflection Group Combinatorics; Funder: National Science Foundation; Code: 1001933
RTG in Combinatorics; Funder: National Science Foundation; Code: 1148634

Bibliographic References

12 Documents citing this article

Alena Ernst;Kai-Uwe Schmidt, 2024, Transitivity in finite general linear groups, Mathematische Zeitschrift, 307, 3, 10.1007/s00209-024-03511-x, https://doi.org/10.1007/s00209-024-03511-x.

Theo Douvropoulos, 2024, Lyashko–Looijenga morphisms and primitive factorizations of Coxeter elements, Mathematische Annalen, 391, 3, pp. 3459-3500, 10.1007/s00208-024-03006-4.

ALENA ERNST;KAI–UWE SCHMIDT, 2023, Intersection theorems for finite general linear groups, Mathematical Proceedings of the Cambridge Philosophical Society, 175, 1, pp. 129-160, 10.1017/s0305004123000075, https://doi.org/10.1017/s0305004123000075.

Graham Gordon, 2023, Cycle type factorizations in GL n 𝔽 q , Algebraic Combinatorics, 5, 6, pp. 1427-1459, 10.5802/alco.259, https://doi.org/10.5802/alco.259.

Theo Douvropoulos, 2023, On enumerating factorizations in reflection groups, Algebraic Combinatorics, 6, 2, pp. 359-385, 10.5802/alco.261, https://doi.org/10.5802/alco.261.

Angèle M. Foley;Alejandro H. Morales;Amarpreet Rattan;Karen Yeats, 2022, Combinatorial and Algebraic Enumeration: a survey of the work of Ian P. Goulden and David M. Jackson, Algebraic Combinatorics, 5, 6, pp. 1205-1226, 10.5802/alco.269, https://doi.org/10.5802/alco.269.

Guillaume Chapuy;Theo Douvropoulos, 2022, Coxeter factorizations with generalized Jucys–Murphy weights and Matrix‐Tree theorems for reflection groups, arXiv (Cornell University), 126, 1, pp. 129-191, 10.1112/plms.12490, http://arxiv.org/abs/2012.04519.

Jia Huang;Joel Brewster Lewis;Victor Reiner;Jia Huang;Joel Brewster Lewis;et al., 2017, Absolute order in general linear groups, arXiv (Cornell University), 95, 1, pp. 223-247, 10.1112/jlms.12013, http://arxiv.org/abs/1506.03332.

Mathieu Guay-Paquet;J. Harnad, 2017, Generating functions for weighted Hurwitz numbers, arXiv (Cornell University), 58, 8, 10.1063/1.4996574, http://arxiv.org/abs/1408.6766.

J. Harnad, 2016, Quantum Hurwitz numbers and Macdonald polynomials, arXiv (Cornell University), 57, 11, 10.1063/1.4967953, http://arxiv.org/abs/1504.03311.

Joel Brewster Lewis;Alejandro H. Morales, 2016, GLn(Fq)-analogues of factorization problems in the symmetric group, arXiv (Cornell University), 58, pp. 75-95, 10.1016/j.ejc.2016.05.004, http://arxiv.org/abs/1511.00711.

Xiaomei Chen;Richard P. Stanley, 2016, A Formula for the Specialization of Skew Schur Functions, Annals of Combinatorics, 20, 3, pp. 539-548, 10.1007/s00026-016-0312-2.

Sources : OpenCitations, OpenAlex & Crossref

Share and export

Consultation statistics

This page has been seen 490 times.

This article's PDF has been downloaded 1192 times.