Gorenstein matrices
Let \(A=(a_{ij})\) be an integral matrix. We say that \(A\) is \((0, 1, 2)\)-matrix if \(a_{ij}\in \{0, 1, 2\}\). There exists the Gorenstein \((0, 1, 2)\)-matrix for any permutation \(\sigma \) on the set \(\{1, \ldots , n\}\) without fixed elements. For every positive integer \(n\) there exists t...
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| Date: | 2018 |
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| Format: | Article |
| Language: | English |
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Lugansk National Taras Shevchenko University
2018
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| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/913 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543173173575680 |
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| author | Dokuchaev, M. A. Kirichenko, V. V. Zelensky, A. V. Zhuravlev, V. N. |
| author_facet | Dokuchaev, M. A. Kirichenko, V. V. Zelensky, A. V. Zhuravlev, V. N. |
| author_sort | Dokuchaev, M. A. |
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| collection | OJS |
| datestamp_date | 2018-03-21T07:18:38Z |
| description | Let \(A=(a_{ij})\) be an integral matrix. We say that \(A\) is \((0, 1, 2)\)-matrix if \(a_{ij}\in \{0, 1, 2\}\). There exists the Gorenstein \((0, 1, 2)\)-matrix for any permutation \(\sigma \) on the set \(\{1, \ldots , n\}\) without fixed elements. For every positive integer \(n\) there exists the Gorenstein cyclic \((0, 1, 2)\)-matrix \(A_{n}\) such that \(inx\,A_{n}=2\).If a Latin square \({\mathcal L}_{n}\) with a first row and first column \((0, 1,\ldots n-1)\) is an exponent matrix, then \(n=2^{m}\) and \({\mathcal L}_{n}\) is the Cayley table of a direct product of \(m\) copies of the cyclic group of order 2. Conversely, the Cayley table \({{\mathcal E}}_{m}\) of the elementary abelian group \(G_{m}=(2)\times\ldots \times (2)\) of order \(2^{m}\) is a Latin square and a Gorenstein symmetric matrix with first row \((0, 1,\ldots , 2^{m}-1)\) and\(\sigma({{\mathcal E}}_{m})=\begin{pmatrix} 1&2&3&\ldots &2^{m}-1&2^{m}\\ 2^{m}&2^{m}-1&2^{m}-2&\ldots & 2&1\end{pmatrix}.\) |
| first_indexed | 2026-02-08T07:59:01Z |
| format | Article |
| id | admjournalluguniveduua-article-913 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2026-02-08T07:59:01Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-9132018-03-21T07:18:38Z Gorenstein matrices Dokuchaev, M. A. Kirichenko, V. V. Zelensky, A. V. Zhuravlev, V. N. exponent matrix; Gorenstein tiled order, Gorenstein matrix, admissible quiver, doubly stochastic matrix 16P40; 16G10 Let \(A=(a_{ij})\) be an integral matrix. We say that \(A\) is \((0, 1, 2)\)-matrix if \(a_{ij}\in \{0, 1, 2\}\). There exists the Gorenstein \((0, 1, 2)\)-matrix for any permutation \(\sigma \) on the set \(\{1, \ldots , n\}\) without fixed elements. For every positive integer \(n\) there exists the Gorenstein cyclic \((0, 1, 2)\)-matrix \(A_{n}\) such that \(inx\,A_{n}=2\).If a Latin square \({\mathcal L}_{n}\) with a first row and first column \((0, 1,\ldots n-1)\) is an exponent matrix, then \(n=2^{m}\) and \({\mathcal L}_{n}\) is the Cayley table of a direct product of \(m\) copies of the cyclic group of order 2. Conversely, the Cayley table \({{\mathcal E}}_{m}\) of the elementary abelian group \(G_{m}=(2)\times\ldots \times (2)\) of order \(2^{m}\) is a Latin square and a Gorenstein symmetric matrix with first row \((0, 1,\ldots , 2^{m}-1)\) and\(\sigma({{\mathcal E}}_{m})=\begin{pmatrix} 1&2&3&\ldots &2^{m}-1&2^{m}\\ 2^{m}&2^{m}-1&2^{m}-2&\ldots & 2&1\end{pmatrix}.\) Lugansk National Taras Shevchenko University 2018-03-21 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/913 Algebra and Discrete Mathematics; Vol 4, No 1 (2005) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/913/442 Copyright (c) 2018 Algebra and Discrete Mathematics |
| spellingShingle | exponent matrix; Gorenstein tiled order Gorenstein matrix admissible quiver doubly stochastic matrix 16P40 16G10 Dokuchaev, M. A. Kirichenko, V. V. Zelensky, A. V. Zhuravlev, V. N. Gorenstein matrices |
| title | Gorenstein matrices |
| title_full | Gorenstein matrices |
| title_fullStr | Gorenstein matrices |
| title_full_unstemmed | Gorenstein matrices |
| title_short | Gorenstein matrices |
| title_sort | gorenstein matrices |
| topic | exponent matrix; Gorenstein tiled order Gorenstein matrix admissible quiver doubly stochastic matrix 16P40 16G10 |
| topic_facet | exponent matrix; Gorenstein tiled order Gorenstein matrix admissible quiver doubly stochastic matrix 16P40 16G10 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/913 |
| work_keys_str_mv | AT dokuchaevma gorensteinmatrices AT kirichenkovv gorensteinmatrices AT zelenskyav gorensteinmatrices AT zhuravlevvn gorensteinmatrices |