Skip to main contentSkip to search
Episciences
Open Access Journals
Sign in(new window)
Discrete Mathematics & Theoretical Computer Science logo
Discrete Mathematics & Theoretical Computer Science
Discrete Mathematics & Theoretical Computer Science logo
Discrete Mathematics & Theoretical Computer Science
Sign in(new window)
Articles & Issues
All articlesAll accepted articlesAll volumesLast volumeSectionsSpecial issuesAuthors
About
The journalIndexing
Boards
Publish
For authorsEthical charter
Submit
Discrete Mathematics & Theoretical Computer Science logo
Contact
|
Credits
eISSN 1365-8050
|
RSS
|
Atom
Episciences
Documentation
|
Acknowledgements
|
Publishing policy
Accessibility: non-compliant
|
Legal mentions
|
Privacy statement
|
Terms of use
  1. Home > Articles & Issues >
  2. Articles >
  3. The freeness of idea ...
Conference paper

The freeness of ideal subarrangements of Weyl arrangements

Takuro Abe (1) (2), Mohamed Barakat ORCID (3), Michael Cuntz ORCID (4), Torsten Hoge (5), Hiroaki Terao (6)
(1) Department of Mechanical Engineering and Science, Kyoto University
(2) Kyōto daigaku = Kyoto University
(3) Fachbereich Mathematik [Kaiserslautern]
(4) Fakultät fur Mathematik und Physik [Hannover]
(5) Fakultät für Mathematik [Bochum]
(6) Department of Mathematics [Sapporo]
Download article
Open on HAL
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.2418
License
https://about.hal.science/hal-authorisation-v1
Indicators
788
Views
547
Downloads

The freeness of ideal subarrangements of Weyl arrangements

Takuro Abe (1) (2), Mohamed Barakat ORCID (3), Michael Cuntz ORCID (4), Torsten Hoge (5), Hiroaki Terao (6)
(1) Department of Mechanical Engineering and Science, Kyoto University
(2) Kyōto daigaku = Kyoto University
(3) Fachbereich Mathematik [Kaiserslautern]
(4) Fakultät fur Mathematik und Physik [Hannover]
(5) Fakultät für Mathematik [Bochum]
(6) Department of Mathematics [Sapporo]
Abstract
English
A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers-Tymoczko. In particular, when an ideal subarrangement is equal to the entire Weyl arrangement, our main theorem yields the celebrated formula by Shapiro, Steinberg, Kostant, and Macdonald. The proof of the main theorem is classification-free. It heavily depends on the theory of free arrangements and thus greatly differs from the earlier proofs of the formula.
French
Un arrangement de Weyl est défini par l’arrangement d’hyperplans du système de racines d’un groupe de Weyl fini. Quand un ensemble de racines positives est un idéal dans le poset de racines, nous appelons l’arrangement correspondant un sous-arrangement idéal. Notre théorème principal affirme que tout sous-arrangement idéal est un arrangement libre et que ses exposants sont donnés par la partition duale de la distribution des hauteurs, ce qui avait été conjecturé par Sommers-Tymoczko. En particulier, quand le sous-arrangement idéal est égal à l’arrangement de Weyl, notre théorème principal donne la célèbre formule par Shapiro, Steinberg, Kostant et Macdonald. La démonstration du théorème principal n’utilise pas de classification. Elle dépend fortement de la théorie des arrangements libres et diffère ainsi grandement des démonstrations précédentes de la formule.
Keywords
  • [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]
  • [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]
  • Weyl arrangement
  • root system
  • ideal
  • free arrangements
  • exponents
  • height
Cited by

Source: OpenCitations

  • MAT-free graphic arrangements and a characterization of strongly chordal graphs by edge-labeling

    Algebraic Combinatorics

    Authors : Tan N. Tran ORCID, Shuhei Tsujie ORCID

    Journal reference : Volume 6, 2024, pp. 1447-1467

    DOI : 10.5802/alco.319

  • Restrictions of aspherical arrangements

    arXiv (Cornell University)

    Authors : Nils Amend, Tilman Möller, Gerhard Röhrle ORCID

    Journal reference : Volume 249, 2018, pp. 67-72

    DOI : 10.1016/j.topol.2018.09.007

  • Arrangements of ideal type

    Journal of Algebra

    Authors : Gerhard Röhrle

    Journal reference : Volume 484, 2017, pp. 126-167

    DOI : 10.1016/j.jalgebra.2017.04.008

  • Addition-deletion theorem for free hyperplane arrangements and combinatorics

    Journal of Algebra

    Authors : Takuro Abe ORCID

    Journal reference : Volume 610, 2022, pp. 1-17

    DOI : 10.1016/j.jalgebra.2022.06.028

  • Characteristic quasi-polynomials of ideals and signed graphs of classical root systems

    European Journal of Combinatorics

    Authors : Tan Nhat Tran ORCID

    Journal reference : Volume 79, 2019, pp. 179-192

    DOI : 10.1016/j.ejc.2019.03.001

  • Accurate arrangements

    arXiv (Cornell University)

    Authors : Paul Mücksch ORCID, Gerhard Röhrle

    Journal reference : Volume 383, 2021, pp. 107702

    DOI : 10.1016/j.aim.2021.107702

  • Derivation degree sequences of non-free arrangements

    arXiv (Cornell University)

    Authors : Max Wakefield

    Journal reference : Volume 47, 2019, pp. 2654-2666

    DOI : 10.1080/00927872.2018.1536207

  • Flag-accurate arrangements

    Innovations in incidence geometry

    Authors : Paul Mücksch ORCID, Gerhard Röhrle, Tan Nhat Tran ORCID

    Journal reference : Volume 21, 2024, pp. 57-116

    DOI : 10.2140/iig.2024.21.57

  • Hyperpolygonal arrangements

    Glasgow Mathematical Journal

    Authors : Lorenzo Giordani, Paul Mücksch ORCID, Gerhard Röhrle, Johannes Schmitt ORCID

    Journal reference : Volume 68, 2025, pp. 134-146

    DOI : 10.1017/s0017089525100633

  • Vines and MAT-labeled graphs

    Forum of Mathematics Sigma

    Authors : Hung Manh Tran ORCID, Tan Nhat Tran ORCID, Shuhei Tsujie ORCID

    Journal reference : Volume 12, 2024, pp.

    DOI : 10.1017/fms.2024.124

  • Arrangements of ideal type are inductively free

    arXiv (Cornell University)

    Authors : Michael Cuntz ORCID, Gerhard Röhrle, Anne Schauenburg

    Journal reference : Volume 29, 2019, pp. 761-773

    DOI : 10.1142/s0218196719500267

  • Inductive Freeness of Ziegler’s Canonical Multiderivations

    Discrete & Computational Geometry

    Authors : Torsten Hoge, Gerhard Röhrle

    Journal reference : Volume 73, 2024, pp. 528-549

    DOI : 10.1007/s00454-024-00644-y

  • Solomon–Terao algebra of hyperplane arrangements

    Journal of the Mathematical Society of Japan

    Authors : Takuro ABE ORCID, Toshiaki MAENO, Satoshi MURAI, Yasuhide NUMATA ORCID

    Journal reference : Volume 71, 2019, pp.

    DOI : 10.2969/jmsj/79957995

  • Worpitzky-compatible subarrangements of braid arrangements and cocomparability graphs

    Comptes Rendus Mathématique

    Authors : Tan Nhat Tran ORCID, Akiyoshi Tsuchiya ORCID

    Journal reference : Volume 359, 2021, pp. 665-674

    DOI : 10.5802/crmath.210

  • Hessenberg varieties and hyperplane arrangements

    Institutional Repositories DataBase (IRDB)

    Authors : Takuro Abe ORCID, Tatsuya Horiguchi ORCID, Mikiya Masuda, Satoshi Murai, Takashi Sato ORCID

    Journal reference : Volume 2020, 2019, pp. 241-286

    DOI : 10.1515/crelle-2018-0039

  • Inductive and divisional posets

    Journal of the London Mathematical Society

    Authors : Roberto Pagaria ORCID, Maddalena Pismataro, Tan Nhat Tran ORCID, Lorenzo Vecchi ORCID

    Journal reference : Volume 109, 2023, pp.

    DOI : 10.1112/jlms.12829

  • Signed graphs and the freeness of the Weyl subarrangements of type Bℓ

    arXiv (Cornell University)

    Authors : Daisuke Suyama ORCID, Michele Torielli ORCID, Shuhei Tsujie ORCID

    Journal reference : Volume 342, 2018, pp. 233-249

    DOI : 10.1016/j.disc.2018.09.029

  • Weight posets associated with gradings of simple Lie algebras, Weyl groups, and arrangements of hyperplanes

    arXiv (Cornell University)

    Authors : Dmitri I. Panyushev ORCID

    Journal reference : Volume 44, 2016, pp. 325-344

    DOI : 10.1007/s10801-016-0671-0

  • $\vee$ -Systems, Holonomy Lie Algebras, and Logarithmic Vector Fields

    International Mathematics Research Notices

    Authors : Misha V Feigin ORCID, Alexander P Veselov ORCID

    Journal reference : Volume 2018, 2016, pp. 2070-2098

    DOI : 10.1093/imrn/rnw289

  • A pattern avoidance criterion for free inversion arrangements

    Journal of Algebraic Combinatorics

    Authors : William Slofstra

    Journal reference : Volume 44, 2016, pp. 201-221

    DOI : 10.1007/s10801-015-0663-5

  • Projective dimension of weakly chordal graphic arrangements

    Algebraic Combinatorics

    Authors : Takuro Abe ORCID, Lukas Kühne ORCID, Paul Mücksch ORCID, Leonie Mühlherr

    Journal reference : Volume 8, 2025, pp. 157-174

    DOI : 10.5802/alco.403

  • On $$A_1^2$$ restrictions of Weyl arrangements

    Journal of Algebraic Combinatorics

    Authors : Takuro Abe, Hiroaki Terao, Tan Nhat Tran ORCID

    Journal reference : Volume 54, 2020, pp. 353-379

    DOI : 10.1007/s10801-020-00979-8

  • Root system chip-firing I: interval-firing

    arXiv (Cornell University)

    Authors : Pavel Galashin, Sam Hopkins, Thomas McConville, Alexander Postnikov

    Journal reference : Volume 292, 2018, pp. 1337-1385

    DOI : 10.1007/s00209-018-2159-1

  • A Survey of Recent Developments on Hessenberg Varieties

    Springer proceedings in mathematics & statistics

    Authors : Hiraku Abe, Tatsuya Horiguchi ORCID

    Journal reference : Volume , 2020, pp. 251-279

    DOI : 10.1007/978-981-15-7451-1_10

  • Plus-one Generated and Next to Free Arrangements of Hyperplanes

    International Mathematics Research Notices

    Authors : Takuro Abe ORCID

    Journal reference : Volume 2021, 2019, pp. 9233-9261

    DOI : 10.1093/imrn/rnz099

  • The topology of arrangements of ideal type

    arXiv (Cornell University)

    Authors : Nils Amend, Gerhard Röhrle

    Journal reference : Volume 19, 2019, pp. 1341-1358

    DOI : 10.2140/agt.2019.19.1341

  • The Largest Coefficient of the Highest Root and the Second Smallest Exponent

    arXiv (Cornell University)

    Authors : Tan Nhat Tran ORCID

    Journal reference : Volume 37, 2020, pp. 127-137

    DOI : 10.1007/s00373-020-02233-1

  • Multiple addition, deletion and restriction theorems for hyperplane arrangements

    Proceedings of the American Mathematical Society

    Authors : Takuro Abe ORCID, Hiroaki Terao ORCID

    Journal reference : Volume 147, 2019, pp. 4835-4845

    DOI : 10.1090/proc/14592
Preview
Loading PDF preview...