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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Lugansk National Taras Shevchenko University
2018
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oai:ojs.admjournal.luguniv.edu.ua: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 |
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Algebra and Discrete Mathematics |
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| datestamp_date |
2018-03-21T07:18:38Z |
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OJS |
| language |
English |
| topic |
exponent matrix; Gorenstein tiled order Gorenstein matrix admissible quiver doubly stochastic matrix 16P40 16G10 |
| 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 |
| topic_facet |
exponent matrix; Gorenstein tiled order Gorenstein matrix admissible quiver doubly stochastic matrix 16P40 16G10 |
| format |
Article |
| 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. |
| title |
Gorenstein matrices |
| title_short |
Gorenstein matrices |
| title_full |
Gorenstein matrices |
| title_fullStr |
Gorenstein matrices |
| title_full_unstemmed |
Gorenstein matrices |
| title_sort |
gorenstein matrices |
| 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\(\sigma({{\mathcal E}}_{m})=\begin{pmatrix} 1&2&3&\ldots &2^{m}-1&2^{m}\\ 2^{m}&2^{m}-1&2^{m}-2&\ldots & 2&1\end{pmatrix}.\) |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/913 |
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AT dokuchaevma gorensteinmatrices AT kirichenkovv gorensteinmatrices AT zelenskyav gorensteinmatrices AT zhuravlevvn gorensteinmatrices |
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2025-07-17T10:31:45Z |
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2025-07-17T10:31:45Z |
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