On Smith normal forms of \(q\)-Varchenko matrices
In this paper, we investigate \(q\)-Varchenko matrices for some hyperplane arrangements with symmetry in two and three dimensions, and prove that they have a Smith normal form over \(\mathbb Z[q]\). In particular, we examine the hyperplane arrangement for the regular \(n\)-gon in the plane and the...
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| Date: | 2023 |
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| Language: | English |
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Lugansk National Taras Shevchenko University
2023
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| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2006 |
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| Journal Title: | Algebra and Discrete Mathematics |
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admjournalluguniveduua-article-20062023-03-06T16:39:17Z On Smith normal forms of \(q\)-Varchenko matrices Boulware, N. Jing, N. Misra, K. C. Smith normal forms, hyperplane arrangements, Platonic solids, Varchenko matrices 15A21; 05E18, 52C35 In this paper, we investigate \(q\)-Varchenko matrices for some hyperplane arrangements with symmetry in two and three dimensions, and prove that they have a Smith normal form over \(\mathbb Z[q]\). In particular, we examine the hyperplane arrangement for the regular \(n\)-gon in the plane and the dihedral model in the space and Platonic polyhedra. In each case, we prove that the \(q\)-Varchenko matrix associated with the hyperplane arrangement has a Smith normal form over \(\mathbb Z[q]\) and realize their congruent transformation matrices over \(\mathbb Z[q]\) as well. Lugansk National Taras Shevchenko University Simons Foundation 2023-03-06 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2006 10.12958/adm2006 Algebra and Discrete Mathematics; Vol 34, No 2 (2022) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2006/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2006/1000 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2006/1012 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2006/1013 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2006/1014 Copyright (c) 2023 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
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| datestamp_date |
2023-03-06T16:39:17Z |
| collection |
OJS |
| language |
English |
| topic |
Smith normal forms hyperplane arrangements Platonic solids Varchenko matrices 15A21; 05E18 52C35 |
| spellingShingle |
Smith normal forms hyperplane arrangements Platonic solids Varchenko matrices 15A21; 05E18 52C35 Boulware, N. Jing, N. Misra, K. C. On Smith normal forms of \(q\)-Varchenko matrices |
| topic_facet |
Smith normal forms hyperplane arrangements Platonic solids Varchenko matrices 15A21; 05E18 52C35 |
| format |
Article |
| author |
Boulware, N. Jing, N. Misra, K. C. |
| author_facet |
Boulware, N. Jing, N. Misra, K. C. |
| author_sort |
Boulware, N. |
| title |
On Smith normal forms of \(q\)-Varchenko matrices |
| title_short |
On Smith normal forms of \(q\)-Varchenko matrices |
| title_full |
On Smith normal forms of \(q\)-Varchenko matrices |
| title_fullStr |
On Smith normal forms of \(q\)-Varchenko matrices |
| title_full_unstemmed |
On Smith normal forms of \(q\)-Varchenko matrices |
| title_sort |
on smith normal forms of \(q\)-varchenko matrices |
| description |
In this paper, we investigate \(q\)-Varchenko matrices for some hyperplane arrangements with symmetry in two and three dimensions, and prove that they have a Smith normal form over \(\mathbb Z[q]\). In particular, we examine the hyperplane arrangement for the regular \(n\)-gon in the plane and the dihedral model in the space and Platonic polyhedra. In each case, we prove that the \(q\)-Varchenko matrix associated with the hyperplane arrangement has a Smith normal form over \(\mathbb Z[q]\) and realize their congruent transformation matrices over \(\mathbb Z[q]\) as well. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2023 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2006 |
| work_keys_str_mv |
AT boulwaren onsmithnormalformsofqvarchenkomatrices AT jingn onsmithnormalformsofqvarchenkomatrices AT misrakc onsmithnormalformsofqvarchenkomatrices |
| first_indexed |
2025-12-02T15:30:46Z |
| last_indexed |
2025-12-02T15:30:46Z |
| _version_ |
1850412059135574016 |