Modular Construction of Free Hyperplane Arrangements
In this article, we study the freeness of hyperplane arrangements. One of the most investigated arrangements is a graphic arrangement. Stanley proved that a graphic arrangement is free if and only if the corresponding graph is chordal, and Dirac showed that a graph is chordal if and only if the grap...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2020 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2020
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/210768 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Modular Construction of Free Hyperplane Arrangements. Shuhei Tsujie. SIGMA 16 (2020), 080, 19 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862695248043442176 |
|---|---|
| author | Tsujie, Shuhei |
| author_facet | Tsujie, Shuhei |
| citation_txt | Modular Construction of Free Hyperplane Arrangements. Shuhei Tsujie. SIGMA 16 (2020), 080, 19 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In this article, we study the freeness of hyperplane arrangements. One of the most investigated arrangements is a graphic arrangement. Stanley proved that a graphic arrangement is free if and only if the corresponding graph is chordal, and Dirac showed that a graph is chordal if and only if the graph is obtained by ''gluing'' complete graphs. We will generalize Dirac's construction to simple matroids with modular joins introduced by Ziegler and show that every arrangement whose associated matroid is constructed in the manner mentioned above is divisionally free. Moreover, we apply the result to arrangements associated with gain graphs and arrangements over finite fields.
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| first_indexed | 2026-03-18T05:19:27Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-210768 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-18T05:19:27Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Tsujie, Shuhei 2025-12-17T14:31:31Z 2020 Modular Construction of Free Hyperplane Arrangements. Shuhei Tsujie. SIGMA 16 (2020), 080, 19 pages 1815-0659 2020 Mathematics Subject Classification: 52C35; 05B35; 05C22; 13N15 arXiv:1908.01535 https://nasplib.isofts.kiev.ua/handle/123456789/210768 https://doi.org/10.3842/SIGMA.2020.080 In this article, we study the freeness of hyperplane arrangements. One of the most investigated arrangements is a graphic arrangement. Stanley proved that a graphic arrangement is free if and only if the corresponding graph is chordal, and Dirac showed that a graph is chordal if and only if the graph is obtained by ''gluing'' complete graphs. We will generalize Dirac's construction to simple matroids with modular joins introduced by Ziegler and show that every arrangement whose associated matroid is constructed in the manner mentioned above is divisionally free. Moreover, we apply the result to arrangements associated with gain graphs and arrangements over finite fields. I greatly appreciate N. Nakashima, D. Suyama, and M. Torielli for valuable discussions, which provide the basis of Section 4. I also owe my deepest gratitude to the anonymous referees whose comments are very helpful in polishing the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Modular Construction of Free Hyperplane Arrangements Article published earlier |
| spellingShingle | Modular Construction of Free Hyperplane Arrangements Tsujie, Shuhei |
| title | Modular Construction of Free Hyperplane Arrangements |
| title_full | Modular Construction of Free Hyperplane Arrangements |
| title_fullStr | Modular Construction of Free Hyperplane Arrangements |
| title_full_unstemmed | Modular Construction of Free Hyperplane Arrangements |
| title_short | Modular Construction of Free Hyperplane Arrangements |
| title_sort | modular construction of free hyperplane arrangements |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210768 |
| work_keys_str_mv | AT tsujieshuhei modularconstructionoffreehyperplanearrangements |