Integer quadratic forms and extensions of subsets of linearly independent roots
We consider subsets of linearly independent roots in a certain root system \(\varPhi\). Let \(S'\) be such a subset, and let \(S'\) be associated with any Carter diagram \(\Gamma'\). The main question of the paper: what root \(\gamma \in \varPhi\) can be added to \(S'\) so that \...
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Lugansk National Taras Shevchenko University
2025
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admjournalluguniveduua-article-23972025-08-13T10:04:13Z Integer quadratic forms and extensions of subsets of linearly independent roots Stekolshchik, Rafael Root system, Cartan matrix, Weyl group, Carater diagram 17B22, 20F55, 20E45 We consider subsets of linearly independent roots in a certain root system \(\varPhi\). Let \(S'\) be such a subset, and let \(S'\) be associated with any Carter diagram \(\Gamma'\). The main question of the paper: what root \(\gamma \in \varPhi\) can be added to \(S'\) so that \(S' \cup \gamma\) is also a subset of linearly independent roots? This extra root \(\gamma\) is called the linkage root. The vector \(\gamma^{\nabla}\) of inner products \(\{(\gamma,\tau'_i)\mid \tau'_i \in S'\}\) is called the linkage label vector. Let \(B_{\Gamma'}\) be the Cartan matrix associated with \(\Gamma'\). It is shown that \(\gamma\) is a linkage root if and only if \(\mathscr{B}^{\vee}_{\Gamma'}(\gamma^{\nabla}) < 2\), where \(\mathscr{B}^{\vee}_{\Gamma'}\) is a quadratic form with the matrix inverse to \(B_{\Gamma'}\). The set of all linkage roots for \(\Gamma'\) is called a linkage system and is denoted by \(\mathscr{L}(\Gamma')\). The sizes of \(\mathscr{L}(\Gamma')\) and \(\mathscr{L}(\Gamma)\) are the same for diagrams \(\Gamma\) and \(\Gamma'\) that have the same rank and \(ADE\) type. Let \(W^{\vee}\) be the Weyl group of the quadratic form \(\mathscr{B}^{\vee}_{\Gamma'}\). The sizes and structure of orbits for linkage systems \(\mathscr{L}(D_l)\) and \(\mathscr{L}(D_l(a_k))\) are presented. Lugansk National Taras Shevchenko University 2025-08-13 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2397 10.12958/adm2397 Algebra and Discrete Mathematics; Vol 39, No 2 (2025) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2397/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2397/1304 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2397/1327 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2397/1328 Copyright (c) 2025 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
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| datestamp_date |
2025-08-13T10:04:13Z |
| collection |
OJS |
| language |
English |
| topic |
Root system Cartan matrix Weyl group Carater diagram 17B22 20F55 20E45 |
| spellingShingle |
Root system Cartan matrix Weyl group Carater diagram 17B22 20F55 20E45 Stekolshchik, Rafael Integer quadratic forms and extensions of subsets of linearly independent roots |
| topic_facet |
Root system Cartan matrix Weyl group Carater diagram 17B22 20F55 20E45 |
| format |
Article |
| author |
Stekolshchik, Rafael |
| author_facet |
Stekolshchik, Rafael |
| author_sort |
Stekolshchik, Rafael |
| title |
Integer quadratic forms and extensions of subsets of linearly independent roots |
| title_short |
Integer quadratic forms and extensions of subsets of linearly independent roots |
| title_full |
Integer quadratic forms and extensions of subsets of linearly independent roots |
| title_fullStr |
Integer quadratic forms and extensions of subsets of linearly independent roots |
| title_full_unstemmed |
Integer quadratic forms and extensions of subsets of linearly independent roots |
| title_sort |
integer quadratic forms and extensions of subsets of linearly independent roots |
| description |
We consider subsets of linearly independent roots in a certain root system \(\varPhi\). Let \(S'\) be such a subset, and let \(S'\) be associated with any Carter diagram \(\Gamma'\). The main question of the paper: what root \(\gamma \in \varPhi\) can be added to \(S'\) so that \(S' \cup \gamma\) is also a subset of linearly independent roots? This extra root \(\gamma\) is called the linkage root. The vector \(\gamma^{\nabla}\) of inner products \(\{(\gamma,\tau'_i)\mid \tau'_i \in S'\}\) is called the linkage label vector. Let \(B_{\Gamma'}\) be the Cartan matrix associated with \(\Gamma'\). It is shown that \(\gamma\) is a linkage root if and only if \(\mathscr{B}^{\vee}_{\Gamma'}(\gamma^{\nabla}) < 2\), where \(\mathscr{B}^{\vee}_{\Gamma'}\) is a quadratic form with the matrix inverse to \(B_{\Gamma'}\). The set of all linkage roots for \(\Gamma'\) is called a linkage system and is denoted by \(\mathscr{L}(\Gamma')\). The sizes of \(\mathscr{L}(\Gamma')\) and \(\mathscr{L}(\Gamma)\) are the same for diagrams \(\Gamma\) and \(\Gamma'\) that have the same rank and \(ADE\) type. Let \(W^{\vee}\) be the Weyl group of the quadratic form \(\mathscr{B}^{\vee}_{\Gamma'}\). The sizes and structure of orbits for linkage systems \(\mathscr{L}(D_l)\) and \(\mathscr{L}(D_l(a_k))\) are presented. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2025 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2397 |
| work_keys_str_mv |
AT stekolshchikrafael integerquadraticformsandextensionsofsubsetsoflinearlyindependentroots |
| first_indexed |
2025-12-02T15:42:43Z |
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2025-12-02T15:42:43Z |
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1850411752719646720 |