Algorithmic computation of principal posets using Maple and Python
We present symbolic and numerical algorithms for a computer search in the Coxeter spectral classification problems. One of the main aims of the paper is to study finite posets \(I\) that are principal, i.e., the rational symmetric Gram matrix \(G_I : = \frac{1}{2}[C_I+ C^{tr}_I]\in\mathbb{M_I(\mathb...
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| Date: | 2018 |
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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/1023 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543330325757952 |
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| author | Gąsiorek, Marcin Simson, Daniel Zając, Katarzyna |
| author_facet | Gąsiorek, Marcin Simson, Daniel Zając, Katarzyna |
| author_sort | Gąsiorek, Marcin |
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| datestamp_date | 2018-04-26T01:41:11Z |
| description | We present symbolic and numerical algorithms for a computer search in the Coxeter spectral classification problems. One of the main aims of the paper is to study finite posets \(I\) that are principal, i.e., the rational symmetric Gram matrix \(G_I : = \frac{1}{2}[C_I+ C^{tr}_I]\in\mathbb{M_I(\mathbb{Q})}\) of \(I\) is positive semi-definite of corank one, where \(C_I\in\mathbb{M}_I(\mathbb{Z})\) is the incidence matrix of \(I\). With any such a connected poset $I$, we associate a simply laced Euclidean diagram \(DI\in \{\widetilde{\mathbb{A}}_n, \widetilde{\mathbb{D}}_n, \widetilde{\mathbb{E}}_6, \widetilde{\mathbb{E}}_7, \widetilde{\mathbb{E}}_8\}\), the Coxeter matrix \(\mbox{ Cox}_I:= - C_I\cdot C^{-tr}_I\), its complex Coxeter spectrum \(\mathbf{specc}_I\), and a reduced Coxeter number \(\check{\mathbf{c}}_I\). One of our aims is to show that the spectrum \(\mathbf{specc}_I\) of any such a poset \(I\) determines the incidence matrix \(C_I\) (hence the poset \(I\)) uniquely, up to a \(\mathbb{Z}\)-congruence. By computer calculations, we find a complete list of principal one-peak posets \(I\) (i.e., \(I\) has a unique maximal element) of cardinality \(\leq 15\), together with \(\mathbf{specc}_I\), \(\check{\mathbf{c}}_I\), the incidence defect \(\partial_I:\mathbb{Z}^I \to\mathbb{Z}\), and the Coxeter-Euclidean type \(DI\). In case when \(DI\in \{\widetilde{\mathbb{A}}_n, \widetilde{\mathbb{D}}_n, \widetilde{\mathbb{E}}_6, \widetilde{\mathbb{E}}_7, \widetilde{\mathbb{E}}_8\}\) and \(n:=|I|\) is relatively small, we show that given such a principal poset \(I\), the incidence matrix \( C_I\) is \(\mathbb{Z}\)-congruent with the non-symmetric Gram matrix \( \check G_{DI}\) of \(DI\), \(\mathbf{specc}_I = \mathbf{specc}_{DI}\) and \(\check{\mathbf{c}} _I= \check{\mathbf{c}}_{DI}\). Moreover, given a pair of principal posets \(I\) and \(J\), with \(|I|= |J| \leq 15\), the matrices \(C_I\) and \(C_J\) are \(\mathbb{Z}\)-congruent if and only if \(\mathbf{specc}_I=\mathbf{specc}_J\). |
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| format | Article |
| id | admjournalluguniveduua-article-1023 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2026-02-08T08:01:30Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
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| spelling | admjournalluguniveduua-article-10232018-04-26T01:41:11Z Algorithmic computation of principal posets using Maple and Python Gąsiorek, Marcin Simson, Daniel Zając, Katarzyna principal poset; edge-bipartite graph; unit quadratic form; computer algorithm; Gram matrix, Coxeter polynomial, Coxeter spectrum 06A11, 15A63, 68R05, 68W30 We present symbolic and numerical algorithms for a computer search in the Coxeter spectral classification problems. One of the main aims of the paper is to study finite posets \(I\) that are principal, i.e., the rational symmetric Gram matrix \(G_I : = \frac{1}{2}[C_I+ C^{tr}_I]\in\mathbb{M_I(\mathbb{Q})}\) of \(I\) is positive semi-definite of corank one, where \(C_I\in\mathbb{M}_I(\mathbb{Z})\) is the incidence matrix of \(I\). With any such a connected poset $I$, we associate a simply laced Euclidean diagram \(DI\in \{\widetilde{\mathbb{A}}_n, \widetilde{\mathbb{D}}_n, \widetilde{\mathbb{E}}_6, \widetilde{\mathbb{E}}_7, \widetilde{\mathbb{E}}_8\}\), the Coxeter matrix \(\mbox{ Cox}_I:= - C_I\cdot C^{-tr}_I\), its complex Coxeter spectrum \(\mathbf{specc}_I\), and a reduced Coxeter number \(\check{\mathbf{c}}_I\). One of our aims is to show that the spectrum \(\mathbf{specc}_I\) of any such a poset \(I\) determines the incidence matrix \(C_I\) (hence the poset \(I\)) uniquely, up to a \(\mathbb{Z}\)-congruence. By computer calculations, we find a complete list of principal one-peak posets \(I\) (i.e., \(I\) has a unique maximal element) of cardinality \(\leq 15\), together with \(\mathbf{specc}_I\), \(\check{\mathbf{c}}_I\), the incidence defect \(\partial_I:\mathbb{Z}^I \to\mathbb{Z}\), and the Coxeter-Euclidean type \(DI\). In case when \(DI\in \{\widetilde{\mathbb{A}}_n, \widetilde{\mathbb{D}}_n, \widetilde{\mathbb{E}}_6, \widetilde{\mathbb{E}}_7, \widetilde{\mathbb{E}}_8\}\) and \(n:=|I|\) is relatively small, we show that given such a principal poset \(I\), the incidence matrix \( C_I\) is \(\mathbb{Z}\)-congruent with the non-symmetric Gram matrix \( \check G_{DI}\) of \(DI\), \(\mathbf{specc}_I = \mathbf{specc}_{DI}\) and \(\check{\mathbf{c}} _I= \check{\mathbf{c}}_{DI}\). Moreover, given a pair of principal posets \(I\) and \(J\), with \(|I|= |J| \leq 15\), the matrices \(C_I\) and \(C_J\) are \(\mathbb{Z}\)-congruent if and only if \(\mathbf{specc}_I=\mathbf{specc}_J\). Lugansk National Taras Shevchenko University 2018-04-26 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1023 Algebra and Discrete Mathematics; Vol 17, No 1 (2014) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1023/547 Copyright (c) 2018 Algebra and Discrete Mathematics |
| spellingShingle | principal poset; edge-bipartite graph; unit quadratic form; computer algorithm; Gram matrix Coxeter polynomial Coxeter spectrum 06A11 15A63 68R05 68W30 Gąsiorek, Marcin Simson, Daniel Zając, Katarzyna Algorithmic computation of principal posets using Maple and Python |
| title | Algorithmic computation of principal posets using Maple and Python |
| title_full | Algorithmic computation of principal posets using Maple and Python |
| title_fullStr | Algorithmic computation of principal posets using Maple and Python |
| title_full_unstemmed | Algorithmic computation of principal posets using Maple and Python |
| title_short | Algorithmic computation of principal posets using Maple and Python |
| title_sort | algorithmic computation of principal posets using maple and python |
| topic | principal poset; edge-bipartite graph; unit quadratic form; computer algorithm; Gram matrix Coxeter polynomial Coxeter spectrum 06A11 15A63 68R05 68W30 |
| topic_facet | principal poset; edge-bipartite graph; unit quadratic form; computer algorithm; Gram matrix Coxeter polynomial Coxeter spectrum 06A11 15A63 68R05 68W30 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1023 |
| work_keys_str_mv | AT gasiorekmarcin algorithmiccomputationofprincipalposetsusingmapleandpython AT simsondaniel algorithmiccomputationofprincipalposetsusingmapleandpython AT zajackatarzyna algorithmiccomputationofprincipalposetsusingmapleandpython |