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Package is "gap-wedderga" Wed Aug 12 16:11:49 2026 rev:6 rq:1370718 version:4.12.0 Changes: -------- --- /work/SRC/openSUSE:Factory/gap-wedderga/gap-wedderga.changes 2025-12-27 11:29:46.789843687 +0100 +++ /work/SRC/openSUSE:Factory/.gap-wedderga.new.17972/gap-wedderga.changes 2026-08-12 16:12:48.791246703 +0200 @@ -1,0 +2,6 @@ +Tue Aug 11 14:31:57 UTC 2026 - Jan Engelhardt <[email protected]> + +- Update to release 4.12.0 + * Updates to the project-internal CI/CD files + +------------------------------------------------------------------- Old: ---- wedderga-4.11.3.tar.gz New: ---- wedderga-4.12.0.tar.gz ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Other differences: ------------------ ++++++ gap-wedderga.spec ++++++ --- /var/tmp/diff_new_pack.LzAsVR/_old 2026-08-12 16:12:49.779288438 +0200 +++ /var/tmp/diff_new_pack.LzAsVR/_new 2026-08-12 16:12:49.779288438 +0200 @@ -1,7 +1,7 @@ # # spec file for package gap-wedderga # -# Copyright (c) 2025 SUSE LLC and contributors +# Copyright (c) 2026 SUSE LLC and contributors # # All modifications and additions to the file contributed by third parties # remain the property of their copyright owners, unless otherwise agreed @@ -17,7 +17,7 @@ Name: gap-wedderga -Version: 4.11.3 +Version: 4.12.0 Release: 0 Summary: GAP: Wedderburn Decomposition of Group Algebras License: GPL-2.0-or-later @@ -28,7 +28,6 @@ BuildArch: noarch BuildRequires: gap-rpm-devel Requires: gap-core >= 4.8 -Requires: gap-gapdoc >= 1.5.1 Suggests: gap-guava >= 3.12 Suggests: gap-laguna >= 3.4 ++++++ _scmsync.obsinfo ++++++ --- /var/tmp/diff_new_pack.LzAsVR/_old 2026-08-12 16:12:49.815289958 +0200 +++ /var/tmp/diff_new_pack.LzAsVR/_new 2026-08-12 16:12:49.819290127 +0200 @@ -1,5 +1,5 @@ -mtime: 1766786549 -commit: 3b23aa1b9c4c44333f4806b535bf7359267ddcef8c168ad5374eb29fad5343bc +mtime: 1786458770 +commit: 057d9cb6e0513ddc7505563a694924ed5b1e1c5af5f65d580da1d9b61a391d60 url: https://src.opensuse.org/jengelh/gap-wedderga revision: master ++++++ build.specials.obscpio ++++++ ++++++ build.specials.obscpio ++++++ diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn' '--exclude=.svnignore' old/.gitignore new/.gitignore --- old/.gitignore 1970-01-01 01:00:00.000000000 +0100 +++ new/.gitignore 2026-08-11 16:32:50.000000000 +0200 @@ -0,0 +1 @@ +.osc ++++++ wedderga-4.11.3.tar.gz -> wedderga-4.12.0.tar.gz ++++++ diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn' '--exclude=.svnignore' old/wedderga-4.11.3/PackageInfo.g new/wedderga-4.12.0/PackageInfo.g --- old/wedderga-4.11.3/PackageInfo.g 2025-12-26 01:15:30.000000000 +0100 +++ new/wedderga-4.12.0/PackageInfo.g 2026-08-11 02:00:00.000000000 +0200 @@ -17,8 +17,8 @@ PackageName := "Wedderga", Subtitle := Concatenation( [ "Wedderburn Decomposition of Group Algebras" ] ), -Version := "4.11.3", -Date := "26/12/2025", # dd/mm/yyyy format +Version := "4.12.0", +Date := "11/08/2026", # dd/mm/yyyy format License := "GPL-2.0-or-later", SourceRepository := rec( @@ -186,7 +186,7 @@ Dependencies := rec( GAP := ">=4.8", - NeededOtherPackages := [ ["GAPDoc", ">= 1.5.1"] ], + NeededOtherPackages := [], SuggestedOtherPackages := [ ["laguna", ">= 3.4"], ["GUAVA", ">= 3.12"] ], ExternalConditions := [] ), diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn' '--exclude=.svnignore' old/wedderga-4.11.3/doc/_entities.xml new/wedderga-4.12.0/doc/_entities.xml --- old/wedderga-4.11.3/doc/_entities.xml 2025-12-26 01:15:30.000000000 +0100 +++ new/wedderga-4.12.0/doc/_entities.xml 2026-08-11 02:00:00.000000000 +0200 @@ -1,4 +1,4 @@ -<!ENTITY VERSION '4.11.3'> -<!ENTITY RELEASEYEAR '2025'> -<!ENTITY RELEASEDATE '26 December 2025'> +<!ENTITY RELEASEDATE '11 August 2026'> +<!ENTITY RELEASEYEAR '2026'> +<!ENTITY VERSION '4.12.0'> <!ENTITY Wedderga '<Package>Wedderga</Package>'> diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn' '--exclude=.svnignore' old/wedderga-4.11.3/doc/chap0.html new/wedderga-4.12.0/doc/chap0.html --- old/wedderga-4.11.3/doc/chap0.html 2025-12-26 01:15:30.000000000 +0100 +++ new/wedderga-4.12.0/doc/chap0.html 2026-08-11 02:00:00.000000000 +0200 @@ -28,9 +28,9 @@ <h2>Wedderburn Decomposition of Group Algebras</h2> -<p>Version 4.11.3</p> +<p>Version 4.12.0</p> -<p>26 December 2025</p> +<p>11 August 2026</p> </div> <p><b>Gurmeet Kaur Bakshi @@ -59,7 +59,7 @@ </b> <br />Email: <span class="URL"><a href="mailto:[email protected]";>[email protected]</a></span> -<br />Homepage: <span class="URL"><a href="https://alex-konovalov.github.io/";>https://alex-konovalov.github.io/</a></span> +<br />Homepage: <span class="URL"><a href="https://olexandr-konovalov.github.io/";>https://olexandr-konovalov.github.io/</a></span> <br />Address: <br />School of Computer Science, University of St Andrews<br /> Jack Cole Building, North Haugh,<br /> St Andrews, Fife, KY16 9SX, Scotland </p><p><b>Sugandha Maheshwary @@ -105,13 +105,13 @@ <p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p> <h3>Copyright</h3> -<p>© 2006-2025 by Gurmeet Kaur Bakshi, Osnel Broche Cristo, Allen Herman, Olexandr Konovalov, Sugandha Maheshwary, Aurora Olivieri, Gabriela Olteanu, Ángel del Río and Inneke Van Gelder.</p> +<p>© 2006-2026 by Gurmeet Kaur Bakshi, Osnel Broche Cristo, Allen Herman, Olexandr Konovalov, Sugandha Maheshwary, Aurora Olivieri, Gabriela Olteanu, Ángel del Río and Inneke Van Gelder.</p> <p><strong class="pkg">Wedderga</strong> is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. For details, see the FSF's own site <span class="URL"><a href="https://www.gnu.org/licenses/gpl.html";>https://www.gnu.org/licenses/gpl.html</a></span>.</p> <p>If you obtained <strong class="pkg">Wedderga</strong>, we would be grateful for a short notification sent to one of the authors. If you publish a result which was partially obtained with the usage of <strong class="pkg">Wedderga</strong>, please cite it in the following form:</p> -<p>G. K. Bakshi, O. Broche Cristo, A. Herman, O. Konovalov, S. Maheshwary, A. Olivieri, G. Olteanu, Á. del Río and I. Van Gelder. <em>Wedderga --- Wedderburn Decomposition of Group Algebras, Version 4.11.3;</em> 2025 (<span class="URL"><a href="https://gap-packages.github.io/wedderga/";>https://gap-packages.github.io/wedderga/</a></span>).</p> +<p>G. K. Bakshi, O. Broche Cristo, A. Herman, O. Konovalov, S. Maheshwary, A. Olivieri, G. Olteanu, Á. del Río and I. Van Gelder. <em>Wedderga --- Wedderburn Decomposition of Group Algebras, Version 4.12.0;</em> 2026 (<span class="URL"><a href="https://gap-packages.github.io/wedderga/";>https://gap-packages.github.io/wedderga/</a></span>).</p> <p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p> <h3>Acknowledgements</h3> diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn' '--exclude=.svnignore' old/wedderga-4.11.3/doc/chap0.txt new/wedderga-4.12.0/doc/chap0.txt --- old/wedderga-4.11.3/doc/chap0.txt 2025-12-26 01:15:30.000000000 +0100 +++ new/wedderga-4.12.0/doc/chap0.txt 2026-08-11 02:00:00.000000000 +0200 @@ -6,10 +6,10 @@ [1XWedderburn Decomposition of Group Algebras[101X - Version 4.11.3 + Version 4.12.0 - 26 December 2025 + 11 August 2026 Gurmeet Kaur Bakshi @@ -56,7 +56,7 @@ Olexandr Konovalov Email: [7Xmailto:[email protected][107X - Homepage: [7Xhttps://alex-konovalov.github.io/[107X + Homepage: [7Xhttps://olexandr-konovalov.github.io/[107X Address: [33X[0;14YSchool of Computer Science, University of St Andrews[133X [33X[0;14YJack Cole Building, North Haugh,[133X [33X[0;14YSt Andrews, Fife, KY16 9SX, Scotland[133X @@ -116,7 +116,7 @@ ------------------------------------------------------- [1XCopyright[101X - [33X[0;0Y© 2006-2025 by Gurmeet Kaur Bakshi, Osnel Broche Cristo, Allen Herman, + [33X[0;0Y© 2006-2026 by Gurmeet Kaur Bakshi, Osnel Broche Cristo, Allen Herman, Olexandr Konovalov, Sugandha Maheshwary, Aurora Olivieri, Gabriela Olteanu, Ángel del Río and Inneke Van Gelder.[133X @@ -132,7 +132,7 @@ [33X[0;0YG. K. Bakshi, O. Broche Cristo, A. Herman, O. Konovalov, S. Maheshwary, A. Olivieri, G. Olteanu, Á. del Río and I. Van Gelder. [13XWedderga --- Wedderburn - Decomposition of Group Algebras, Version 4.11.3;[113X 2025 + Decomposition of Group Algebras, Version 4.12.0;[113X 2026 ([7Xhttps://gap-packages.github.io/wedderga/[107X).[133X diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn' '--exclude=.svnignore' old/wedderga-4.11.3/doc/chap0_mj.html new/wedderga-4.12.0/doc/chap0_mj.html --- old/wedderga-4.11.3/doc/chap0_mj.html 2025-12-26 01:15:30.000000000 +0100 +++ new/wedderga-4.12.0/doc/chap0_mj.html 2026-08-11 02:00:00.000000000 +0200 @@ -31,9 +31,9 @@ <h2>Wedderburn Decomposition of Group Algebras</h2> -<p>Version 4.11.3</p> +<p>Version 4.12.0</p> -<p>26 December 2025</p> +<p>11 August 2026</p> </div> <p><b>Gurmeet Kaur Bakshi @@ -62,7 +62,7 @@ </b> <br />Email: <span class="URL"><a href="mailto:[email protected]";>[email protected]</a></span> -<br />Homepage: <span class="URL"><a href="https://alex-konovalov.github.io/";>https://alex-konovalov.github.io/</a></span> +<br />Homepage: <span class="URL"><a href="https://olexandr-konovalov.github.io/";>https://olexandr-konovalov.github.io/</a></span> <br />Address: <br />School of Computer Science, University of St Andrews<br /> Jack Cole Building, North Haugh,<br /> St Andrews, Fife, KY16 9SX, Scotland </p><p><b>Sugandha Maheshwary @@ -108,13 +108,13 @@ <p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p> <h3>Copyright</h3> -<p>© 2006-2025 by Gurmeet Kaur Bakshi, Osnel Broche Cristo, Allen Herman, Olexandr Konovalov, Sugandha Maheshwary, Aurora Olivieri, Gabriela Olteanu, Ángel del Río and Inneke Van Gelder.</p> +<p>© 2006-2026 by Gurmeet Kaur Bakshi, Osnel Broche Cristo, Allen Herman, Olexandr Konovalov, Sugandha Maheshwary, Aurora Olivieri, Gabriela Olteanu, Ángel del Río and Inneke Van Gelder.</p> <p><strong class="pkg">Wedderga</strong> is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. For details, see the FSF's own site <span class="URL"><a href="https://www.gnu.org/licenses/gpl.html";>https://www.gnu.org/licenses/gpl.html</a></span>.</p> <p>If you obtained <strong class="pkg">Wedderga</strong>, we would be grateful for a short notification sent to one of the authors. If you publish a result which was partially obtained with the usage of <strong class="pkg">Wedderga</strong>, please cite it in the following form:</p> -<p>G. K. Bakshi, O. Broche Cristo, A. Herman, O. Konovalov, S. Maheshwary, A. Olivieri, G. Olteanu, Á. del Río and I. Van Gelder. <em>Wedderga --- Wedderburn Decomposition of Group Algebras, Version 4.11.3;</em> 2025 (<span class="URL"><a href="https://gap-packages.github.io/wedderga/";>https://gap-packages.github.io/wedderga/</a></span>).</p> +<p>G. K. Bakshi, O. Broche Cristo, A. Herman, O. Konovalov, S. Maheshwary, A. Olivieri, G. Olteanu, Á. del Río and I. Van Gelder. <em>Wedderga --- Wedderburn Decomposition of Group Algebras, Version 4.12.0;</em> 2026 (<span class="URL"><a href="https://gap-packages.github.io/wedderga/";>https://gap-packages.github.io/wedderga/</a></span>).</p> <p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p> <h3>Acknowledgements</h3> diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn' '--exclude=.svnignore' old/wedderga-4.11.3/doc/chap1.html new/wedderga-4.12.0/doc/chap1.html --- old/wedderga-4.11.3/doc/chap1.html 2025-12-26 01:15:30.000000000 +0100 +++ new/wedderga-4.12.0/doc/chap1.html 2026-08-11 02:00:00.000000000 +0200 @@ -93,7 +93,7 @@ <p><code class="func">WedderburnDecomposition</code> (<a href="chap2.html#X7F1779ED8777F3E7"><span class="RefLink">2.1-1</span></a>) computes a list of simple algebras such that their direct product is isomorphic to the group algebra <span class="SimpleMath">FG</span>, given as input. Thus, the direct product of the entries of the output is the <em>Wedderburn decomposition</em> (<a href="chap9.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of <span class="SimpleMath">FG</span>.</p> -<p>If <span class="SimpleMath">F</span> is an abelian number field then the entries of the output are given as matrix algebras over cyclotomic algebras (see <a href="chap9.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>), thus, the entries of the output of <code class="func">WedderburnDecomposition</code> (<a href="chap2.html#X7F1779ED8777F3E7"><span class="RefLink">2.1-1</span></a>) are realizations of the <em>Wedderburn components</em> (<a href="chap9.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of <span class="SimpleMath">FG</span> as algebras which are <em>Brauer equivalent</em> (<a href="chap9.html#X7A24D5407F72C633"><span class="RefLink">9.5</span></a>) to <em>cyclotomic algebras</em> (<a href="chap9.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>). Recall that the Brauer-Witt Theorem ensures that every simple factor of a semisimple group ring <span class="SimpleMath">FG</span> is Brauer equivalent (that is represents the same class in the Brauer group of its centre) to a cyclotomic algebra (<a href="chapBib.html#biBY">[Yam74]</a>. In this case the algorithm is based on a computational oriented proof of the Brauer-Witt Theorem due to Olteanu <a href="chapBib.html#biBO">[Olt07]</a> which uses previous work by Olivieri, del Río and Simón <a href="chapBib.html#biBORS">[OdRS04]</a> (see also <a href="chapBib.html#biBOR">[OdR03]</a> ) for rational group algebras of <em>strongly monomial groups</em> (<a href="chap9.html#X84C694978557EFE5"><span class="RefLink">9.17</span></a>). The algorithms are also based upon the work of Bakshi and Maheshwary <a href="chapBib.html#biBBM14">[BM14]</a> (see also <a href="chapBib.html#biBBM16">[BM16]</a>) on the rational group algebras of <em>normally monomial groups</em> (<a href="chap9.html#X7C8D47C180E0ACAD"><span class="RefLink">9.18</span></a>).</p> +<p>If <span class="SimpleMath">F</span> is an abelian number field then the entries of the output are given as matrix algebras over cyclotomic algebras (see <a href="chap9.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>), thus, the entries of the output of <code class="func">WedderburnDecomposition</code> (<a href="chap2.html#X7F1779ED8777F3E7"><span class="RefLink">2.1-1</span></a>) are realizations of the <em>Wedderburn components</em> (<a href="chap9.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of <span class="SimpleMath">FG</span> as algebras which are <em>Brauer equivalent</em> (<a href="chap9.html#X7A24D5407F72C633"><span class="RefLink">9.5</span></a>) to <em>cyclotomic algebras</em> (<a href="chap9.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>). Recall that the Brauer-Witt Theorem ensures that every simple factor of a semisimple group ring <span class="SimpleMath">FG</span> is Brauer equivalent (that is represents the same class in the Brauer group of its centre) to a cyclotomic algebra (<a href="chapBib.html#biBY">[Yam74]</a>. In this case the algorithm is based on a computational oriented proof of the Brauer-Witt Theorem due to Olteanu <a href="chapBib.html#biBO">[Olt07]</a> which uses previous work by Olivieri, del Río and Simón <a href="chapBib.html#biBORS">[OdS04]</a> (see also <a href="chapBib.html#biBOR">[Od03]</a> ) for rational group algebras of <em>strongly monomial groups</em> (<a href="chap9.html#X84C694978557EFE5"><span class="RefLink">9.17</span></a>). The algorithms are also based upon the work of Bakshi and Maheshwary <a href="chapBib.html#biBBM14">[BM14]</a> (see also <a href="chapBib.html#biBBM16">[BM16]</a>) on the rational group algebras of <em>normally monomial groups</em> (<a href="chap9.html#X7C8D47C180E0ACAD"><span class="RefLink">9.18</span></a>).</p> <p>The Wedderburn components of <span class="SimpleMath">FG</span> are also matrix algebras over division rings which are finite extensions of the field <span class="SimpleMath">F</span>. If <span class="SimpleMath">F</span> is finite then by the Wedderburn theorem these division rings are finite fields. In this case the output of <code class="func">WedderburnDecomposition</code> (<a href="chap2.html#X7F1779ED8777F3E7"><span class="RefLink">2.1-1</span></a>) represents the factors of <span class="SimpleMath">FG</span> as matrix algebras over finite extensions of the field <span class="SimpleMath">F</span>.</p> diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn' '--exclude=.svnignore' old/wedderga-4.11.3/doc/chap1.txt new/wedderga-4.12.0/doc/chap1.txt --- old/wedderga-4.11.3/doc/chap1.txt 2025-12-26 01:15:30.000000000 +0100 +++ new/wedderga-4.12.0/doc/chap1.txt 2026-08-11 02:00:00.000000000 +0200 @@ -91,10 +91,10 @@ centre) to a cyclotomic algebra ([Yam74]. In this case the algorithm is based on a computational oriented proof of the Brauer-Witt Theorem due to Olteanu [Olt07] which uses previous work by Olivieri, del Río and Simón - [OdRS04] (see also [OdR03] ) for rational group algebras of [13Xstrongly - monomial groups[113X ([14X9.17[114X). The algorithms are also based upon the work of - Bakshi and Maheshwary [BM14] (see also [BM16]) on the rational group - algebras of [13Xnormally monomial groups[113X ([14X9.18[114X).[133X + [OdS04] (see also [Od03] ) for rational group algebras of [13Xstrongly monomial + groups[113X ([14X9.17[114X). The algorithms are also based upon the work of Bakshi and + Maheshwary [BM14] (see also [BM16]) on the rational group algebras of + [13Xnormally monomial groups[113X ([14X9.18[114X).[133X [33X[0;0YThe Wedderburn components of [22XFG[122X are also matrix algebras over division rings which are finite extensions of the field [22XF[122X. If [22XF[122X is finite then by the diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn' '--exclude=.svnignore' old/wedderga-4.11.3/doc/chap1_mj.html new/wedderga-4.12.0/doc/chap1_mj.html --- old/wedderga-4.11.3/doc/chap1_mj.html 2025-12-26 01:15:30.000000000 +0100 +++ new/wedderga-4.12.0/doc/chap1_mj.html 2026-08-11 02:00:00.000000000 +0200 @@ -96,7 +96,7 @@ <p><code class="func">WedderburnDecomposition</code> (<a href="chap2_mj.html#X7F1779ED8777F3E7"><span class="RefLink">2.1-1</span></a>) computes a list of simple algebras such that their direct product is isomorphic to the group algebra <span class="SimpleMath">\(FG\)</span>, given as input. Thus, the direct product of the entries of the output is the <em>Wedderburn decomposition</em> (<a href="chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of <span class="SimpleMath">\(FG\)</span>.</p> -<p>If <span class="SimpleMath">\(F\)</span> is an abelian number field then the entries of the output are given as matrix algebras over cyclotomic algebras (see <a href="chap9_mj.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>), thus, the entries of the output of <code class="func">WedderburnDecomposition</code> (<a href="chap2_mj.html#X7F1779ED8777F3E7"><span class="RefLink">2.1-1</span></a>) are realizations of the <em>Wedderburn components</em> (<a href="chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of <span class="SimpleMath">\(FG\)</span> as algebras which are <em>Brauer equivalent</em> (<a href="chap9_mj.html#X7A24D5407F72C633"><span class="RefLink">9.5</span></a>) to <em>cyclotomic algebras</em> (<a href="chap9_mj.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>). Recall that the Brauer-Witt Theorem ensures that every simple factor of a semisimple group ring <span class="SimpleMath">\(FG\)</span> is Brauer equivalent (that i s represents the same class in the Brauer group of its centre) to a cyclotomic algebra (<a href="chapBib_mj.html#biBY">[Yam74]</a>. In this case the algorithm is based on a computational oriented proof of the Brauer-Witt Theorem due to Olteanu <a href="chapBib_mj.html#biBO">[Olt07]</a> which uses previous work by Olivieri, del Río and Simón <a href="chapBib_mj.html#biBORS">[OdRS04]</a> (see also <a href="chapBib_mj.html#biBOR">[OdR03]</a> ) for rational group algebras of <em>strongly monomial groups</em> (<a href="chap9_mj.html#X84C694978557EFE5"><span class="RefLink">9.17</span></a>). The algorithms are also based upon the work of Bakshi and Maheshwary <a href="chapBib_mj.html#biBBM14">[BM14]</a> (see also <a href="chapBib_mj.html#biBBM16">[BM16]</a>) on the rational group algebras of <em>normally monomial groups</em> (<a href="chap9_mj.html#X7C8D47C180E0ACAD"><span class="RefLink">9.18</span></a>).</p> +<p>If <span class="SimpleMath">\(F\)</span> is an abelian number field then the entries of the output are given as matrix algebras over cyclotomic algebras (see <a href="chap9_mj.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>), thus, the entries of the output of <code class="func">WedderburnDecomposition</code> (<a href="chap2_mj.html#X7F1779ED8777F3E7"><span class="RefLink">2.1-1</span></a>) are realizations of the <em>Wedderburn components</em> (<a href="chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of <span class="SimpleMath">\(FG\)</span> as algebras which are <em>Brauer equivalent</em> (<a href="chap9_mj.html#X7A24D5407F72C633"><span class="RefLink">9.5</span></a>) to <em>cyclotomic algebras</em> (<a href="chap9_mj.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>). Recall that the Brauer-Witt Theorem ensures that every simple factor of a semisimple group ring <span class="SimpleMath">\(FG\)</span> is Brauer equivalent (that i s represents the same class in the Brauer group of its centre) to a cyclotomic algebra (<a href="chapBib_mj.html#biBY">[Yam74]</a>. In this case the algorithm is based on a computational oriented proof of the Brauer-Witt Theorem due to Olteanu <a href="chapBib_mj.html#biBO">[Olt07]</a> which uses previous work by Olivieri, del Río and Simón <a href="chapBib_mj.html#biBORS">[OdS04]</a> (see also <a href="chapBib_mj.html#biBOR">[Od03]</a> ) for rational group algebras of <em>strongly monomial groups</em> (<a href="chap9_mj.html#X84C694978557EFE5"><span class="RefLink">9.17</span></a>). The algorithms are also based upon the work of Bakshi and Maheshwary <a href="chapBib_mj.html#biBBM14">[BM14]</a> (see also <a href="chapBib_mj.html#biBBM16">[BM16]</a>) on the rational group algebras of <em>normally monomial groups</em> (<a href="chap9_mj.html#X7C8D47C180E0ACAD"><span class="RefLink">9.18</span></a>).</p> <p>The Wedderburn components of <span class="SimpleMath">\(FG\)</span> are also matrix algebras over division rings which are finite extensions of the field <span class="SimpleMath">\(F\)</span>. If <span class="SimpleMath">\(F\)</span> is finite then by the Wedderburn theorem these division rings are finite fields. In this case the output of <code class="func">WedderburnDecomposition</code> (<a href="chap2_mj.html#X7F1779ED8777F3E7"><span class="RefLink">2.1-1</span></a>) represents the factors of <span class="SimpleMath">\(FG\)</span> as matrix algebras over finite extensions of the field <span class="SimpleMath">\(F\)</span>.</p> diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn' '--exclude=.svnignore' old/wedderga-4.11.3/doc/chap9.html new/wedderga-4.12.0/doc/chap9.html --- old/wedderga-4.11.3/doc/chap9.html 2025-12-26 01:15:30.000000000 +0100 +++ new/wedderga-4.12.0/doc/chap9.html 2026-08-11 02:00:00.000000000 +0200 @@ -155,7 +155,7 @@ <p>In some cases (for the zero characteristic coefficient field) the size <span class="SimpleMath">n</span> of the matrix algebras is not a positive integer but a positive rational number. This is a consequence of the fact that the <em>Brauer-Witt Theorem</em> <a href="chapBib.html#biBY">[Yam74]</a> only ensures that each <em>Wedderburn component</em> (<a href="chap9.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of a semisimple group algebra is Brauer equivalent (<a href="chap9.html#X7A24D5407F72C633"><span class="RefLink">9.5</span></a>) to a <em>cyclotomic algebra</em> (<a href="chap9.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>), but not necessarily isomorphic to a full matrix algebra of a cyclotomic algebra. For example, a Wedderburn component <span class="SimpleMath">D</span> of a group algebra can be a division algebra but not a cyclotomic algebra. In this case <span class="SimpleMath">M_n(D)</span> is a cyclotomic algebra <span class="SimpleMa th">C</span> for some <span class="SimpleMath">n</span> and therefore <span class="SimpleMath">D</span> can be described as <span class="SimpleMath">M_1/n(C)</span> (see last Example in <code class="func">WedderburnDecomposition</code> (<a href="chap2.html#X7F1779ED8777F3E7"><span class="RefLink">2.1-1</span></a>)).</p> -<p>The main algorithm of <strong class="pkg">Wedderga</strong> is based on a computational oriented proof of the Brauer-Witt Theorem due to Olteanu <a href="chapBib.html#biBO">[Olt07]</a> which uses previous work by Olivieri, del Río and Simón <a href="chapBib.html#biBORS">[OdRS04]</a> (see also <a href="chapBib.html#biBOR">[OdR03]</a> ) for rational group algebras of <em>strongly monomial groups</em> (<a href="chap9.html#X84C694978557EFE5"><span class="RefLink">9.17</span></a>). The algorithms are also based upon the work of Bakshi and Maheshwary <a href="chapBib.html#biBBM14">[BM14]</a> (see also <a href="chapBib.html#biBBM16">[BM16]</a>) on the rational group algebras of <em>normally monomial groups</em> (<a href="chap9.html#X7C8D47C180E0ACAD"><span class="RefLink">9.18</span></a>).</p> +<p>The main algorithm of <strong class="pkg">Wedderga</strong> is based on a computational oriented proof of the Brauer-Witt Theorem due to Olteanu <a href="chapBib.html#biBO">[Olt07]</a> which uses previous work by Olivieri, del Río and Simón <a href="chapBib.html#biBORS">[OdS04]</a> (see also <a href="chapBib.html#biBOR">[Od03]</a> ) for rational group algebras of <em>strongly monomial groups</em> (<a href="chap9.html#X84C694978557EFE5"><span class="RefLink">9.17</span></a>). The algorithms are also based upon the work of Bakshi and Maheshwary <a href="chapBib.html#biBBM14">[BM14]</a> (see also <a href="chapBib.html#biBBM16">[BM16]</a>) on the rational group algebras of <em>normally monomial groups</em> (<a href="chap9.html#X7C8D47C180E0ACAD"><span class="RefLink">9.18</span></a>).</p> <p><a id="X87B6505C7C2EE054" name="X87B6505C7C2EE054"></a></p> @@ -450,7 +450,7 @@ <h4>9.14 <span class="Heading">Shoda pairs of a group</span></h4> -<p>Let <span class="SimpleMath">G</span> be a finite group. A <em>Shoda pair</em> of <span class="SimpleMath">G</span> is a pair <span class="SimpleMath">(K,H)</span> of subgroups of <span class="SimpleMath">G</span> for which there is a linear character <span class="SimpleMath">χ</span> of <span class="SimpleMath">K</span> with kernel <span class="SimpleMath">H</span> such that the induced character <span class="SimpleMath">χ^G</span> in <span class="SimpleMath">G</span> is irreducible. By <a href="chapBib.html#biBS">[Sho33]</a> or <a href="chapBib.html#biBORS">[OdRS04]</a>, <span class="SimpleMath">(K,H)</span> is a Shoda pair if and only if the following conditions hold:</p> +<p>Let <span class="SimpleMath">G</span> be a finite group. A <em>Shoda pair</em> of <span class="SimpleMath">G</span> is a pair <span class="SimpleMath">(K,H)</span> of subgroups of <span class="SimpleMath">G</span> for which there is a linear character <span class="SimpleMath">χ</span> of <span class="SimpleMath">K</span> with kernel <span class="SimpleMath">H</span> such that the induced character <span class="SimpleMath">χ^G</span> in <span class="SimpleMath">G</span> is irreducible. By <a href="chapBib.html#biBS">[Sho33]</a> or <a href="chapBib.html#biBORS">[OdS04]</a>, <span class="SimpleMath">(K,H)</span> is a Shoda pair if and only if the following conditions hold:</p> <p>* <span class="SimpleMath">H</span> is normal in <span class="SimpleMath">K</span>,</p> @@ -458,7 +458,7 @@ <p>* if <span class="SimpleMath">K^g ∩ K ⊆ H</span> for some <span class="SimpleMath">g ∈ G</span> then <span class="SimpleMath">g ∈ K</span>.</p> -<p>If <span class="SimpleMath">(K,H)</span> is a Shoda pair and <span class="SimpleMath">χ</span> is a linear character of <span class="SimpleMath">K≤ G</span> with kernel <span class="SimpleMath">H</span> then the <em>primitive central idempotent</em> (<a href="chap9.html#X87B6505C7C2EE054"><span class="RefLink">9.4</span></a>) of <span class="SimpleMath">ℚ G</span> associated to the irreducible character <span class="SimpleMath">χ^G</span> is of the form <span class="SimpleMath">e=e_ℚ (χ^G)=a e(G,K,H)</span> for some <span class="SimpleMath">a ∈ ℚ</span> <a href="chapBib.html#biBORS">[OdRS04]</a> (see <a href="chap9.html#X8310E96086509397"><span class="RefLink">9.13</span></a> for the definition of <span class="SimpleMath">e(G,K,H)</span>). In that case we say that <span class="SimpleMath">e</span> is the <em>primitive central idempotent realized by the Shoda pair</em> <span class="SimpleMath">(K,H)</span> of <span class="SimpleMath">G</span>.</p> +<p>If <span class="SimpleMath">(K,H)</span> is a Shoda pair and <span class="SimpleMath">χ</span> is a linear character of <span class="SimpleMath">K≤ G</span> with kernel <span class="SimpleMath">H</span> then the <em>primitive central idempotent</em> (<a href="chap9.html#X87B6505C7C2EE054"><span class="RefLink">9.4</span></a>) of <span class="SimpleMath">ℚ G</span> associated to the irreducible character <span class="SimpleMath">χ^G</span> is of the form <span class="SimpleMath">e=e_ℚ (χ^G)=a e(G,K,H)</span> for some <span class="SimpleMath">a ∈ ℚ</span> <a href="chapBib.html#biBORS">[OdS04]</a> (see <a href="chap9.html#X8310E96086509397"><span class="RefLink">9.13</span></a> for the definition of <span class="SimpleMath">e(G,K,H)</span>). In that case we say that <span class="SimpleMath">e</span> is the <em>primitive central idempotent realized by the Shoda pair</em> <span class="SimpleMath">(K,H)</span> of <span class="SimpleMath">G</span>.</p> <p>A group <span class="SimpleMath">G</span> is monomial, that is every irreducible character of <span class="SimpleMath">G</span> is monomial, if and only if every primitive central idempotent of <span class="SimpleMath">ℚ G</span> is realizable by a Shoda pair of <span class="SimpleMath">G</span>.</p> @@ -474,11 +474,11 @@ <p>* for every <span class="SimpleMath">g ∈ G∖ N</span> , <span class="SimpleMath">ε(K,H)ε(K,H)^g=0</span>. (See <a href="chap9.html#X8310E96086509397"><span class="RefLink">9.13</span></a> for the definition of <span class="SimpleMath">ε(K,H)</span>).</p> -<p>Let <span class="SimpleMath">(K,H)</span> be a strong Shoda pair of <span class="SimpleMath">G</span>. Then <span class="SimpleMath">(K,H)</span> is a Shoda pair (<a href="chap9.html#X7D518BAB80EDE190"><span class="RefLink">9.14</span></a>) of <span class="SimpleMath">G</span>. Thus there is a linear character <span class="SimpleMath">θ</span> of <span class="SimpleMath">K</span> with kernel <span class="SimpleMath">H</span> such that the induced character <span class="SimpleMath">χ=χ(G,K,H)=θ^G</span> is irreducible. Moreover the <em>primitive central idempotent</em> (<a href="chap9.html#X87B6505C7C2EE054"><span class="RefLink">9.4</span></a>) <span class="SimpleMath">e_ℚ (χ)</span> of <span class="SimpleMath">ℚ G</span> realized by <span class="SimpleMath">(K,H)</span> is <span class="SimpleMath">e(G,K,H)</span>, see <a href="chapBib.html#biBORS">[OdRS04]</a>.</p> +<p>Let <span class="SimpleMath">(K,H)</span> be a strong Shoda pair of <span class="SimpleMath">G</span>. Then <span class="SimpleMath">(K,H)</span> is a Shoda pair (<a href="chap9.html#X7D518BAB80EDE190"><span class="RefLink">9.14</span></a>) of <span class="SimpleMath">G</span>. Thus there is a linear character <span class="SimpleMath">θ</span> of <span class="SimpleMath">K</span> with kernel <span class="SimpleMath">H</span> such that the induced character <span class="SimpleMath">χ=χ(G,K,H)=θ^G</span> is irreducible. Moreover the <em>primitive central idempotent</em> (<a href="chap9.html#X87B6505C7C2EE054"><span class="RefLink">9.4</span></a>) <span class="SimpleMath">e_ℚ (χ)</span> of <span class="SimpleMath">ℚ G</span> realized by <span class="SimpleMath">(K,H)</span> is <span class="SimpleMath">e(G,K,H)</span>, see <a href="chapBib.html#biBORS">[OdS04]</a>.</p> <p>Two <em>strong Shoda pairs</em> (<a href="chap9.html#X7E3479527BAE5B9E"><span class="RefLink">9.15</span></a>) <span class="SimpleMath">(K_1,H_1)</span> and <span class="SimpleMath">(K_2,H_2)</span> of <span class="SimpleMath">G</span> are said to be <em>equivalent</em> if the characters <span class="SimpleMath">χ(G,K_1,H_1)</span> and <span class="SimpleMath">χ(G,K_2,H_2)</span> are Galois conjugate, or equivalently if <span class="SimpleMath">e(G,K_1,H_1)=e(G,K_2,H_2)</span>. A set of representatives of strong Shoda pairs of <span class="SimpleMath">G</span> is termed as a complete irredundant set of strong Shoda pairs of <span class="SimpleMath">G</span>.</p> -<p>The advantage of strong Shoda pairs over Shoda pairs is that one can describe the simple algebra <span class="SimpleMath">FGe_F(χ)</span> as a matrix algebra of a <em>cyclotomic algebra</em> (<a href="chap9.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>, see <a href="chapBib.html#biBORS">[OdRS04]</a> for <span class="SimpleMath">F=ℚ</span> and <a href="chapBib.html#biBO">[Olt07]</a> for the general case).</p> +<p>The advantage of strong Shoda pairs over Shoda pairs is that one can describe the simple algebra <span class="SimpleMath">FGe_F(χ)</span> as a matrix algebra of a <em>cyclotomic algebra</em> (<a href="chap9.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>, see <a href="chapBib.html#biBORS">[OdS04]</a> for <span class="SimpleMath">F=ℚ</span> and <a href="chapBib.html#biBO">[Olt07]</a> for the general case).</p> <p>More precisely, <span class="SimpleMath">ℚ Ge(G,K,H)</span> is isomorphic to <span class="SimpleMath">M_n(ℚ (ξ)*_a^t N/K)</span>, where <span class="SimpleMath">ξ</span> is a <span class="SimpleMath">[K:H]</span>-th root of unity, <span class="SimpleMath">N</span> is the normalizer of <span class="SimpleMath">H</span> in <span class="SimpleMath">G</span>, <span class="SimpleMath">n=[G:N]</span> and <span class="SimpleMath">ℚ (ξ)*_a^t N/K</span> is a <em>crossed product</em> (see <a href="chap9.html#X7FB21779832CE1CB"><span class="RefLink">9.6</span></a>) with action <span class="SimpleMath">a</span> and twisting <span class="SimpleMath">t</span> given as follows:</p> @@ -494,7 +494,7 @@ t(r,s) = \xi^j, \mbox{ if } \varphi(rs)^{-1} \varphi(r)\varphi(s) = x^j, </p> -<p>for <span class="SimpleMath">r,s ∈ N/K</span> and integers <span class="SimpleMath">i</span> and <span class="SimpleMath">j</span>, see <a href="chapBib.html#biBORS">[OdRS04]</a>. Notice that the cocycle is the one given by the natural extension</p> +<p>for <span class="SimpleMath">r,s ∈ N/K</span> and integers <span class="SimpleMath">i</span> and <span class="SimpleMath">j</span>, see <a href="chapBib.html#biBORS">[OdS04]</a>. Notice that the cocycle is the one given by the natural extension</p> <p class="pcenter"> 1 \rightarrow K/H \rightarrow N/H \rightarrow N/K \rightarrow 1 @@ -528,7 +528,7 @@ <p>The group <span class="SimpleMath">G</span> is <em>strongly monomial</em> if every irreducible character of <span class="SimpleMath">G</span> is strongly monomial.</p> -<p>Strong Shoda pairs where firstly introduced by Olivieri, del Río and Simón who proved that every abelian-by-supersolvable group is strongly monomial <a href="chapBib.html#biBORS">[OdRS04]</a>. The algorithm to compute the Wedderburn decomposition of rational group algebras for strongly monomial groups was explained in <a href="chapBib.html#biBOR">[OdR03]</a>. This method was extended for semisimple finite group algebras by Broche Cristo and del Río in <a href="chapBib.html#biBBR">[BdR07]</a> (see Section <a href="chap9.html#X800D8C5087D79DC8"><span class="RefLink">9.19</span></a>). Finally, Olteanu <a href="chapBib.html#biBO">[Olt07]</a> shows how to compute the <em>Wedderburn decomposition</em> (<a href="chap9.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of an arbitrary semisimple group ring by making use of not only the strong Shoda pairs of <span class="SimpleMath">G</span> but also the strong Shoda pairs of the subgroups of <span class="SimpleMath">G</span >.</p> +<p>Strong Shoda pairs where firstly introduced by Olivieri, del Río and Simón who proved that every abelian-by-supersolvable group is strongly monomial <a href="chapBib.html#biBORS">[OdS04]</a>. The algorithm to compute the Wedderburn decomposition of rational group algebras for strongly monomial groups was explained in <a href="chapBib.html#biBOR">[Od03]</a>. This method was extended for semisimple finite group algebras by Broche Cristo and del Río in <a href="chapBib.html#biBBR">[Bd07]</a> (see Section <a href="chap9.html#X800D8C5087D79DC8"><span class="RefLink">9.19</span></a>). Finally, Olteanu <a href="chapBib.html#biBO">[Olt07]</a> shows how to compute the <em>Wedderburn decomposition</em> (<a href="chap9.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of an arbitrary semisimple group ring by making use of not only the strong Shoda pairs of <span class="SimpleMath">G</span> but also the strong Shoda pairs of the subgroups of <span class="SimpleMath">G</span>.< /p> <p><a id="X7C8D47C180E0ACAD" name="X7C8D47C180E0ACAD"></a></p> @@ -566,9 +566,9 @@ <p>where <span class="SimpleMath">c</span> is an arbitrary element of <span class="SimpleMath">C</span> and <span class="SimpleMath">tr</span> is the trace map of the field extension <span class="SimpleMath">F(ξ)/F</span>. Then <span class="SimpleMath">ε_C(K,H)</span> does not depend on the choice of <span class="SimpleMath">c ∈ C</span> and is a <em>primitive central idempotent</em> (<a href="chap9.html#X87B6505C7C2EE054"><span class="RefLink">9.4</span></a>) of <span class="SimpleMath">FK</span>.</p> -<p>Finally, let <span class="SimpleMath">e_C(G,K,H)</span> denote the sum of the different <span class="SimpleMath">G</span>-conjugates of <span class="SimpleMath">ε_C(K,H)</span>. Then <span class="SimpleMath">e_C(G,K,H)</span> is a <em>primitive central idempotent</em> (<a href="chap9.html#X87B6505C7C2EE054"><span class="RefLink">9.4</span></a>) of <span class="SimpleMath">FG</span> <a href="chapBib.html#biBBR">[BdR07]</a>. We say that <span class="SimpleMath">e_C(G,K,H)</span> is the primitive central idempotent realized by the strong Shoda pair <span class="SimpleMath">(K,H)</span> of the group <span class="SimpleMath">G</span> and the cyclotomic class <span class="SimpleMath">C</span>.</p> +<p>Finally, let <span class="SimpleMath">e_C(G,K,H)</span> denote the sum of the different <span class="SimpleMath">G</span>-conjugates of <span class="SimpleMath">ε_C(K,H)</span>. Then <span class="SimpleMath">e_C(G,K,H)</span> is a <em>primitive central idempotent</em> (<a href="chap9.html#X87B6505C7C2EE054"><span class="RefLink">9.4</span></a>) of <span class="SimpleMath">FG</span> <a href="chapBib.html#biBBR">[Bd07]</a>. We say that <span class="SimpleMath">e_C(G,K,H)</span> is the primitive central idempotent realized by the strong Shoda pair <span class="SimpleMath">(K,H)</span> of the group <span class="SimpleMath">G</span> and the cyclotomic class <span class="SimpleMath">C</span>.</p> -<p>If <span class="SimpleMath">G</span> is <em>strongly monomial</em> (<a href="chap9.html#X84C694978557EFE5"><span class="RefLink">9.17</span></a>) then every primitive central idempotent of <span class="SimpleMath">FG</span> is realizable by some <em>strong Shoda pair</em> (<a href="chap9.html#X7E3479527BAE5B9E"><span class="RefLink">9.15</span></a>) of <span class="SimpleMath">G</span> and some cyclotomic class <span class="SimpleMath">C</span> <a href="chapBib.html#biBBR">[BdR07]</a>. As in the zero characteristic case, this explain how to compute the <em>Wedderburn decomposition</em> (<a href="chap9.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of <span class="SimpleMath">FG</span> for a finite semisimple algebra of a strongly monomial group (see <a href="chapBib.html#biBBR">[BdR07]</a> for details). For non strongly monomial groups the algorithm to compute the Wedderburn decomposition just uses the Brauer characters.</p> +<p>If <span class="SimpleMath">G</span> is <em>strongly monomial</em> (<a href="chap9.html#X84C694978557EFE5"><span class="RefLink">9.17</span></a>) then every primitive central idempotent of <span class="SimpleMath">FG</span> is realizable by some <em>strong Shoda pair</em> (<a href="chap9.html#X7E3479527BAE5B9E"><span class="RefLink">9.15</span></a>) of <span class="SimpleMath">G</span> and some cyclotomic class <span class="SimpleMath">C</span> <a href="chapBib.html#biBBR">[Bd07]</a>. As in the zero characteristic case, this explain how to compute the <em>Wedderburn decomposition</em> (<a href="chap9.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of <span class="SimpleMath">FG</span> for a finite semisimple algebra of a strongly monomial group (see <a href="chapBib.html#biBBR">[Bd07]</a> for details). For non strongly monomial groups the algorithm to compute the Wedderburn decomposition just uses the Brauer characters.</p> <p>.</p> @@ -602,7 +602,7 @@ <p>When <span class="SimpleMath">R</span> is a semisimple ring, then every left ideal <span class="SimpleMath">L</span> of <span class="SimpleMath">R</span> is of the form <span class="SimpleMath">L=Re</span>, where <span class="SimpleMath">e</span> is an idempotent of <span class="SimpleMath">R</span>. Therefore, we can use the idempotents to characterize the decompositions of semisimple rings as a direct sum of minimal left ideals. In particular, let <span class="SimpleMath">R=⊕_i=1^t L_i</span> be a decomposition of a semisimple ring as a direct sum of minimal left ideals. Then, there exists a family <span class="SimpleMath">{e_1,dots,e_t}</span> of elements of <span class="SimpleMath">R</span> such that: each <span class="SimpleMath">e_i≠ 0</span> is an idempotent element, if <span class="SimpleMath">i≠ j</span>, then <span class="SimpleMath">e_ie_j=0</span>, <span class="SimpleMath">1=e_1+⋯+e_t</span> and each <span class="SimpleMath">e_i</span> cannot be written as <sp an class="SimpleMath">e_i=e_i'+e_i''</span>, where <span class="SimpleMath">e_i',e_i''</span> are idempotents such that <span class="SimpleMath">e_i',e_i''≠ 0</span> and <span class="SimpleMath">e_i'e_i''=0</span>, <span class="SimpleMath">1≤ i≤</span>. Conversely, if there exists a family of idempotents <span class="SimpleMath">{e_1,dots,e_t}</span> satisfying the previous conditions, then the left ideals <span class="SimpleMath">L_i=Re_i</span> are minimal and <span class="SimpleMath">R=⊕_i=1^t L_i</span>. Such a set of idempotents is called a <em>complete set of orthogonal primitive idempotents</em> of the ring <span class="SimpleMath">R</span>. Such a set is not uniquely determined.</p> -<p>Let <span class="SimpleMath">F</span> be a finite field and <span class="SimpleMath">G</span> a finite nilpotent group such that <span class="SimpleMath">F G</span> is semisimple. Let <span class="SimpleMath">(H,K)</span> be a strong Shoda pair of <span class="SimpleMath">G</span>, <span class="SimpleMath">C∈mathcalC(H/K)</span> and set <span class="SimpleMath">e_C=e_C(G,H,K)</span>, <span class="SimpleMath">ε_C=ε_C(H,K)</span>, <span class="SimpleMath">H/K=⟨overlinea⟩</span>, <span class="SimpleMath">E=E_G(H/K)</span>. Let <span class="SimpleMath">E_2/K</span> and <span class="SimpleMath">H_2/K=⟨overlinea_2⟩</span> (respectively <span class="SimpleMath">E_2'/K</span> and <span class="SimpleMath">H_2'/K=⟨overlinea_2'}⟩</span>) denote the 2-parts (respectively 2'-parts) of <span class="SimpleMath">E/K</span> and <span class="SimpleMath">H/K</span> respectively. Then <span class="SimpleMath">⟨overlinea_2'}⟩</span> has a cyclic complement <span class="SimpleMath" >⟨overlineb_2'}⟩</span> in <span class="SimpleMath">E_2'/K</span>. Using the >description of the primitive central idempotents and the Wedderburn >components of a semisimple finite group algebra <span class="SimpleMath">F >G</span> (<a href="chap9.html#X800D8C5087D79DC8"><span >class="RefLink">9.19</span></a>), a complete set of orthogonal primitive >idempotents of <span class="SimpleMath">F Ge_C</span> is described (see <a >href="chapBib.html#biBOV">[OVG11]</a>) as the set of conjugates of <span >class="SimpleMath">β_e_C=widetildeb_2'}β_2ε_C</span> by the elements of <span >class="SimpleMath">T_e_C=T_2'T_2T_E</span>, where <span >class="SimpleMath">T_2'={1,a_2',a_2'^2,dots,a_2'^[E_2':H_2']-1}</span>, <span >class="SimpleMath">T_E</span> denotes a right transversal of <span >class="SimpleMath">E</span> in <span class="SimpleMath">G</span> and <span >class="SimpleMath">β_2</span> and <span class="SimpleMath">T_2</span> are >given according to the cases below.</p> +<p>Let <span class="SimpleMath">F</span> be a finite field and <span class="SimpleMath">G</span> a finite nilpotent group such that <span class="SimpleMath">F G</span> is semisimple. Let <span class="SimpleMath">(H,K)</span> be a strong Shoda pair of <span class="SimpleMath">G</span>, <span class="SimpleMath">C∈mathcalC(H/K)</span> and set <span class="SimpleMath">e_C=e_C(G,H,K)</span>, <span class="SimpleMath">ε_C=ε_C(H,K)</span>, <span class="SimpleMath">H/K=⟨overlinea⟩</span>, <span class="SimpleMath">E=E_G(H/K)</span>. Let <span class="SimpleMath">E_2/K</span> and <span class="SimpleMath">H_2/K=⟨overlinea_2⟩</span> (respectively <span class="SimpleMath">E_2'/K</span> and <span class="SimpleMath">H_2'/K=⟨overlinea_2'}⟩</span>) denote the 2-parts (respectively 2'-parts) of <span class="SimpleMath">E/K</span> and <span class="SimpleMath">H/K</span> respectively. Then <span class="SimpleMath">⟨overlinea_2'}⟩</span> has a cyclic complement <span class="SimpleMath" >⟨overlineb_2'}⟩</span> in <span class="SimpleMath">E_2'/K</span>. Using the >description of the primitive central idempotents and the Wedderburn >components of a semisimple finite group algebra <span class="SimpleMath">F >G</span> (<a href="chap9.html#X800D8C5087D79DC8"><span >class="RefLink">9.19</span></a>), a complete set of orthogonal primitive >idempotents of <span class="SimpleMath">F Ge_C</span> is described (see <a >href="chapBib.html#biBOV">[OV11]</a>) as the set of conjugates of <span >class="SimpleMath">β_e_C=widetildeb_2'}β_2ε_C</span> by the elements of <span >class="SimpleMath">T_e_C=T_2'T_2T_E</span>, where <span >class="SimpleMath">T_2'={1,a_2',a_2'^2,dots,a_2'^[E_2':H_2']-1}</span>, <span >class="SimpleMath">T_E</span> denotes a right transversal of <span >class="SimpleMath">E</span> in <span class="SimpleMath">G</span> and <span >class="SimpleMath">β_2</span> and <span class="SimpleMath">T_2</span> are >given according to the cases below.</p> <ol> <li><p>If <span class="SimpleMath">H_2/K</span> has a complement <span class="SimpleMath">M_2/K</span> in <span class="SimpleMath">E_2/K</span> then <span class="SimpleMath">β_2=widetildeM_2</span>. Moreover, if <span class="SimpleMath">M_2/K</span> is cyclic, then there exists <span class="SimpleMath">b_2∈ E_2</span> such that <span class="SimpleMath">E_2/K</span> is given by the following presentation</p> @@ -671,7 +671,7 @@ \{x\widehat{T_1}\varepsilon x^{-1} \mid x\in T_2\langle{x_e}\rangle\} </p> -<p>is a complete set of orthogonal primitive idempotents of <span class="SimpleMath">F G e</span> where <span class="SimpleMath">x_e=ψ^-1(PAP^-1)</span>, <span class="SimpleMath">T_1</span> is a transversal of <span class="SimpleMath">H</span> in <span class="SimpleMath">E</span> and <span class="SimpleMath">T_2</span> is a right transversal of <span class="SimpleMath">E</span> in <span class="SimpleMath">G</span> (<a href="chapBib.html#biBOV2">[OVG15]</a>). By <span class="SimpleMath">widehatT_1</span> we denote the element <span class="SimpleMath">frac1|T_1|∑_t∈ T_1t</span> in <span class="SimpleMath">F G</span>.</p> +<p>is a complete set of orthogonal primitive idempotents of <span class="SimpleMath">F G e</span> where <span class="SimpleMath">x_e=ψ^-1(PAP^-1)</span>, <span class="SimpleMath">T_1</span> is a transversal of <span class="SimpleMath">H</span> in <span class="SimpleMath">E</span> and <span class="SimpleMath">T_2</span> is a right transversal of <span class="SimpleMath">E</span> in <span class="SimpleMath">G</span> (<a href="chapBib.html#biBOV2">[OV15]</a>). By <span class="SimpleMath">widehatT_1</span> we denote the element <span class="SimpleMath">frac1|T_1|∑_t∈ T_1t</span> in <span class="SimpleMath">F G</span>.</p> <p><a id="X856D7975810BF987" name="X856D7975810BF987"></a></p> diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn' '--exclude=.svnignore' old/wedderga-4.11.3/doc/chap9.txt new/wedderga-4.12.0/doc/chap9.txt --- old/wedderga-4.11.3/doc/chap9.txt 2025-12-26 01:15:30.000000000 +0100 +++ new/wedderga-4.12.0/doc/chap9.txt 2026-08-11 02:00:00.000000000 +0200 @@ -107,7 +107,7 @@ [33X[0;0YThe main algorithm of [5XWedderga[105X is based on a computational oriented proof of the Brauer-Witt Theorem due to Olteanu [Olt07] which uses previous work by - Olivieri, del Río and Simón [OdRS04] (see also [OdR03] ) for rational group + Olivieri, del Río and Simón [OdS04] (see also [Od03] ) for rational group algebras of [13Xstrongly monomial groups[113X ([14X9.17[114X). The algorithms are also based upon the work of Bakshi and Maheshwary [BM14] (see also [BM16]) on the rational group algebras of [13Xnormally monomial groups[113X ([14X9.18[114X).[133X @@ -504,8 +504,8 @@ [33X[0;0YLet [22XG[122X be a finite group. A [13XShoda pair[113X of [22XG[122X is a pair [22X(K,H)[122X of subgroups of [22XG[122X for which there is a linear character [22Xχ[122X of [22XK[122X with kernel [22XH[122X such that the - induced character [22Xχ^G[122X in [22XG[122X is irreducible. By [Sho33] or [OdRS04], [22X(K,H)[122X is - a Shoda pair if and only if the following conditions hold:[133X + induced character [22Xχ^G[122X in [22XG[122X is irreducible. By [Sho33] or [OdS04], [22X(K,H)[122X is a + Shoda pair if and only if the following conditions hold:[133X [33X[0;0Y* [22XH[122X is normal in [22XK[122X,[133X @@ -516,7 +516,7 @@ [33X[0;0YIf [22X(K,H)[122X is a Shoda pair and [22Xχ[122X is a linear character of [22XK≤ G[122X with kernel [22XH[122X then the [13Xprimitive central idempotent[113X ([14X9.4[114X) of [22Xℚ G[122X associated to the irreducible character [22Xχ^G[122X is of the form [22Xe=e_ℚ (χ^G)=a e(G,K,H)[122X for some [22Xa ∈ - ℚ[122X [OdRS04] (see [14X9.13[114X for the definition of [22Xe(G,K,H)[122X). In that case we say + ℚ[122X [OdS04] (see [14X9.13[114X for the definition of [22Xe(G,K,H)[122X). In that case we say that [22Xe[122X is the [13Xprimitive central idempotent realized by the Shoda pair[113X [22X(K,H)[122X of [22XG[122X.[133X @@ -541,7 +541,7 @@ [22XG[122X. Thus there is a linear character [22Xθ[122X of [22XK[122X with kernel [22XH[122X such that the induced character [22Xχ=χ(G,K,H)=θ^G[122X is irreducible. Moreover the [13Xprimitive central idempotent[113X ([14X9.4[114X) [22Xe_ℚ (χ)[122X of [22Xℚ G[122X realized by [22X(K,H)[122X is [22Xe(G,K,H)[122X, see - [OdRS04].[133X + [OdS04].[133X [33X[0;0YTwo [13Xstrong Shoda pairs[113X ([14X9.15[114X) [22X(K_1,H_1)[122X and [22X(K_2,H_2)[122X of [22XG[122X are said to be [13Xequivalent[113X if the characters [22Xχ(G,K_1,H_1)[122X and [22Xχ(G,K_2,H_2)[122X are Galois @@ -551,7 +551,7 @@ [33X[0;0YThe advantage of strong Shoda pairs over Shoda pairs is that one can describe the simple algebra [22XFGe_F(χ)[122X as a matrix algebra of a [13Xcyclotomic - algebra[113X ([14X9.11[114X, see [OdRS04] for [22XF=ℚ[122X and [Olt07] for the general case).[133X + algebra[113X ([14X9.11[114X, see [OdS04] for [22XF=ℚ[122X and [Olt07] for the general case).[133X [33X[0;0YMore precisely, [22Xℚ Ge(G,K,H)[122X is isomorphic to [22XM_n(ℚ (ξ)*_a^t N/K)[122X, where [22Xξ[122X is a [22X[K:H][122X-th root of unity, [22XN[122X is the normalizer of [22XH[122X in [22XG[122X, [22Xn=[G:N][122X and [22Xℚ @@ -573,7 +573,7 @@ [124X - [33X[0;0Yfor [22Xr,s ∈ N/K[122X and integers [22Xi[122X and [22Xj[122X, see [OdRS04]. Notice that the cocycle is + [33X[0;0Yfor [22Xr,s ∈ N/K[122X and integers [22Xi[122X and [22Xj[122X, see [OdS04]. Notice that the cocycle is the one given by the natural extension[133X @@ -632,11 +632,11 @@ [33X[0;0YStrong Shoda pairs where firstly introduced by Olivieri, del Río and Simón who proved that every abelian-by-supersolvable group is strongly monomial - [OdRS04]. The algorithm to compute the Wedderburn decomposition of rational - group algebras for strongly monomial groups was explained in [OdR03]. This + [OdS04]. The algorithm to compute the Wedderburn decomposition of rational + group algebras for strongly monomial groups was explained in [Od03]. This method was extended for semisimple finite group algebras by Broche Cristo - and del Río in [BdR07] (see Section [14X9.19[114X). Finally, Olteanu [Olt07] shows - how to compute the [13XWedderburn decomposition[113X ([14X9.3[114X) of an arbitrary semisimple + and del Río in [Bd07] (see Section [14X9.19[114X). Finally, Olteanu [Olt07] shows how + to compute the [13XWedderburn decomposition[113X ([14X9.3[114X) of an arbitrary semisimple group ring by making use of not only the strong Shoda pairs of [22XG[122X but also the strong Shoda pairs of the subgroups of [22XG[122X.[133X @@ -694,14 +694,14 @@ [33X[0;0YFinally, let [22Xe_C(G,K,H)[122X denote the sum of the different [22XG[122X-conjugates of [22Xε_C(K,H)[122X. Then [22Xe_C(G,K,H)[122X is a [13Xprimitive central idempotent[113X ([14X9.4[114X) of [22XFG[122X - [BdR07]. We say that [22Xe_C(G,K,H)[122X is the primitive central idempotent realized + [Bd07]. We say that [22Xe_C(G,K,H)[122X is the primitive central idempotent realized by the strong Shoda pair [22X(K,H)[122X of the group [22XG[122X and the cyclotomic class [22XC[122X.[133X [33X[0;0YIf [22XG[122X is [13Xstrongly monomial[113X ([14X9.17[114X) then every primitive central idempotent of [22XFG[122X is realizable by some [13Xstrong Shoda pair[113X ([14X9.15[114X) of [22XG[122X and some cyclotomic - class [22XC[122X [BdR07]. As in the zero characteristic case, this explain how to + class [22XC[122X [Bd07]. As in the zero characteristic case, this explain how to compute the [13XWedderburn decomposition[113X ([14X9.3[114X) of [22XFG[122X for a finite semisimple - algebra of a strongly monomial group (see [BdR07] for details). For non + algebra of a strongly monomial group (see [Bd07] for details). For non strongly monomial groups the algorithm to compute the Wedderburn decomposition just uses the Brauer characters.[133X @@ -859,7 +859,7 @@ [22X⟨overlinea_2'}⟩[122X has a cyclic complement [22X⟨overlineb_2'}⟩[122X in [22XE_2'/K[122X. Using the description of the primitive central idempotents and the Wedderburn components of a semisimple finite group algebra [22XF G[122X ([14X9.19[114X), a complete set - of orthogonal primitive idempotents of [22XF Ge_C[122X is described (see [OVG11]) as + of orthogonal primitive idempotents of [22XF Ge_C[122X is described (see [OV11]) as the set of conjugates of [22Xβ_e_C=widetildeb_2'}β_2ε_C[122X by the elements of [22XT_e_C=T_2'T_2T_E[122X, where [22XT_2'={1,a_2',a_2'^2,dots,a_2'^[E_2':H_2']-1}[122X, [22XT_E[122X denotes a right transversal of [22XE[122X in [22XG[122X and [22Xβ_2[122X and [22XT_2[122X are given according to @@ -955,7 +955,7 @@ [33X[0;0Yis a complete set of orthogonal primitive idempotents of [22XF G e[122X where [22Xx_e=ψ^-1(PAP^-1)[122X, [22XT_1[122X is a transversal of [22XH[122X in [22XE[122X and [22XT_2[122X is a right - transversal of [22XE[122X in [22XG[122X ([OVG15]). By [22XwidehatT_1[122X we denote the element + transversal of [22XE[122X in [22XG[122X ([OV15]). By [22XwidehatT_1[122X we denote the element [22Xfrac1|T_1|∑_t∈ T_1t[122X in [22XF G[122X.[133X diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn' '--exclude=.svnignore' old/wedderga-4.11.3/doc/chap9_mj.html new/wedderga-4.12.0/doc/chap9_mj.html --- old/wedderga-4.11.3/doc/chap9_mj.html 2025-12-26 01:15:30.000000000 +0100 +++ new/wedderga-4.12.0/doc/chap9_mj.html 2026-08-11 02:00:00.000000000 +0200 @@ -158,7 +158,7 @@ <p>In some cases (for the zero characteristic coefficient field) the size <span class="SimpleMath">\(n\)</span> of the matrix algebras is not a positive integer but a positive rational number. This is a consequence of the fact that the <em>Brauer-Witt Theorem</em> <a href="chapBib_mj.html#biBY">[Yam74]</a> only ensures that each <em>Wedderburn component</em> (<a href="chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of a semisimple group algebra is Brauer equivalent (<a href="chap9_mj.html#X7A24D5407F72C633"><span class="RefLink">9.5</span></a>) to a <em>cyclotomic algebra</em> (<a href="chap9_mj.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>), but not necessarily isomorphic to a full matrix algebra of a cyclotomic algebra. For example, a Wedderburn component <span class="SimpleMath">\(D\)</span> of a group algebra can be a division algebra but not a cyclotomic algebra. In this case <span class="SimpleMath">\(M_n(D)\)</span> is a cyclotomic algeb ra <span class="SimpleMath">\(C\)</span> for some <span class="SimpleMath">\(n\)</span> and therefore <span class="SimpleMath">\(D\)</span> can be described as <span class="SimpleMath">\(M_{1/n}(C)\)</span> (see last Example in <code class="func">WedderburnDecomposition</code> (<a href="chap2_mj.html#X7F1779ED8777F3E7"><span class="RefLink">2.1-1</span></a>)).</p> -<p>The main algorithm of <strong class="pkg">Wedderga</strong> is based on a computational oriented proof of the Brauer-Witt Theorem due to Olteanu <a href="chapBib_mj.html#biBO">[Olt07]</a> which uses previous work by Olivieri, del Río and Simón <a href="chapBib_mj.html#biBORS">[OdRS04]</a> (see also <a href="chapBib_mj.html#biBOR">[OdR03]</a> ) for rational group algebras of <em>strongly monomial groups</em> (<a href="chap9_mj.html#X84C694978557EFE5"><span class="RefLink">9.17</span></a>). The algorithms are also based upon the work of Bakshi and Maheshwary <a href="chapBib_mj.html#biBBM14">[BM14]</a> (see also <a href="chapBib_mj.html#biBBM16">[BM16]</a>) on the rational group algebras of <em>normally monomial groups</em> (<a href="chap9_mj.html#X7C8D47C180E0ACAD"><span class="RefLink">9.18</span></a>).</p> +<p>The main algorithm of <strong class="pkg">Wedderga</strong> is based on a computational oriented proof of the Brauer-Witt Theorem due to Olteanu <a href="chapBib_mj.html#biBO">[Olt07]</a> which uses previous work by Olivieri, del Río and Simón <a href="chapBib_mj.html#biBORS">[OdS04]</a> (see also <a href="chapBib_mj.html#biBOR">[Od03]</a> ) for rational group algebras of <em>strongly monomial groups</em> (<a href="chap9_mj.html#X84C694978557EFE5"><span class="RefLink">9.17</span></a>). The algorithms are also based upon the work of Bakshi and Maheshwary <a href="chapBib_mj.html#biBBM14">[BM14]</a> (see also <a href="chapBib_mj.html#biBBM16">[BM16]</a>) on the rational group algebras of <em>normally monomial groups</em> (<a href="chap9_mj.html#X7C8D47C180E0ACAD"><span class="RefLink">9.18</span></a>).</p> <p><a id="X87B6505C7C2EE054" name="X87B6505C7C2EE054"></a></p> @@ -453,7 +453,7 @@ <h4>9.14 <span class="Heading">Shoda pairs of a group</span></h4> -<p>Let <span class="SimpleMath">\(G\)</span> be a finite group. A <em>Shoda pair</em> of <span class="SimpleMath">\(G\)</span> is a pair <span class="SimpleMath">\((K,H)\)</span> of subgroups of <span class="SimpleMath">\(G\)</span> for which there is a linear character <span class="SimpleMath">\(\chi\)</span> of <span class="SimpleMath">\(K\)</span> with kernel <span class="SimpleMath">\(H\)</span> such that the induced character <span class="SimpleMath">\(\chi^G\)</span> in <span class="SimpleMath">\(G\)</span> is irreducible. By <a href="chapBib_mj.html#biBS">[Sho33]</a> or <a href="chapBib_mj.html#biBORS">[OdRS04]</a>, <span class="SimpleMath">\((K,H)\)</span> is a Shoda pair if and only if the following conditions hold:</p> +<p>Let <span class="SimpleMath">\(G\)</span> be a finite group. A <em>Shoda pair</em> of <span class="SimpleMath">\(G\)</span> is a pair <span class="SimpleMath">\((K,H)\)</span> of subgroups of <span class="SimpleMath">\(G\)</span> for which there is a linear character <span class="SimpleMath">\(\chi\)</span> of <span class="SimpleMath">\(K\)</span> with kernel <span class="SimpleMath">\(H\)</span> such that the induced character <span class="SimpleMath">\(\chi^G\)</span> in <span class="SimpleMath">\(G\)</span> is irreducible. By <a href="chapBib_mj.html#biBS">[Sho33]</a> or <a href="chapBib_mj.html#biBORS">[OdS04]</a>, <span class="SimpleMath">\((K,H)\)</span> is a Shoda pair if and only if the following conditions hold:</p> <p>* <span class="SimpleMath">\(H\)</span> is normal in <span class="SimpleMath">\(K\)</span>,</p> @@ -461,7 +461,7 @@ <p>* if <span class="SimpleMath">\(K^g \cap K \subseteq H\)</span> for some <span class="SimpleMath">\(g \in G\)</span> then <span class="SimpleMath">\(g \in K\)</span>.</p> -<p>If <span class="SimpleMath">\((K,H)\)</span> is a Shoda pair and <span class="SimpleMath">\(\chi\)</span> is a linear character of <span class="SimpleMath">\(K\le G\)</span> with kernel <span class="SimpleMath">\(H\)</span> then the <em>primitive central idempotent</em> (<a href="chap9_mj.html#X87B6505C7C2EE054"><span class="RefLink">9.4</span></a>) of <span class="SimpleMath">\(ℚ G\)</span> associated to the irreducible character <span class="SimpleMath">\(\chi^G\)</span> is of the form <span class="SimpleMath">\(e=e_ℚ (\chi^G)=a e(G,K,H)\)</span> for some <span class="SimpleMath">\(a \in ℚ \)</span> <a href="chapBib_mj.html#biBORS">[OdRS04]</a> (see <a href="chap9_mj.html#X8310E96086509397"><span class="RefLink">9.13</span></a> for the definition of <span class="SimpleMath">\(e(G,K,H)\)</span>). In that case we say that <span class="SimpleMath">\(e\)</span> is the <em>primitive central idempotent realized by the Shoda pair</em> <span class="SimpleMath">\((K,H)\)</span> of <span class="SimpleMath">\(G\)</span>.</p> +<p>If <span class="SimpleMath">\((K,H)\)</span> is a Shoda pair and <span class="SimpleMath">\(\chi\)</span> is a linear character of <span class="SimpleMath">\(K\le G\)</span> with kernel <span class="SimpleMath">\(H\)</span> then the <em>primitive central idempotent</em> (<a href="chap9_mj.html#X87B6505C7C2EE054"><span class="RefLink">9.4</span></a>) of <span class="SimpleMath">\(ℚ G\)</span> associated to the irreducible character <span class="SimpleMath">\(\chi^G\)</span> is of the form <span class="SimpleMath">\(e=e_ℚ (\chi^G)=a e(G,K,H)\)</span> for some <span class="SimpleMath">\(a \in ℚ \)</span> <a href="chapBib_mj.html#biBORS">[OdS04]</a> (see <a href="chap9_mj.html#X8310E96086509397"><span class="RefLink">9.13</span></a> for the definition of <span class="SimpleMath">\(e(G,K,H)\)</span>). In that case we say that <span class="SimpleMath">\(e\)</span> is the <em>primitive central idempotent realized by the Shoda pair</em> <span class="SimpleMath">\((K,H)\)</span> of <span class="SimpleMath">\(G\)</span>.</p> <p>A group <span class="SimpleMath">\(G\)</span> is monomial, that is every irreducible character of <span class="SimpleMath">\(G\)</span> is monomial, if and only if every primitive central idempotent of <span class="SimpleMath">\(ℚ G\)</span> is realizable by a Shoda pair of <span class="SimpleMath">\(G\)</span>.</p> @@ -477,11 +477,11 @@ <p>* for every <span class="SimpleMath">\(g \in G\setminus N\)</span> , <span class="SimpleMath">\(\varepsilon(K,H)\varepsilon(K,H)^g=0\)</span>. (See <a href="chap9_mj.html#X8310E96086509397"><span class="RefLink">9.13</span></a> for the definition of <span class="SimpleMath">\(\varepsilon(K,H)\)</span>).</p> -<p>Let <span class="SimpleMath">\((K,H)\)</span> be a strong Shoda pair of <span class="SimpleMath">\(G\)</span>. Then <span class="SimpleMath">\((K,H)\)</span> is a Shoda pair (<a href="chap9_mj.html#X7D518BAB80EDE190"><span class="RefLink">9.14</span></a>) of <span class="SimpleMath">\(G\)</span>. Thus there is a linear character <span class="SimpleMath">\(\theta\)</span> of <span class="SimpleMath">\(K\)</span> with kernel <span class="SimpleMath">\(H\)</span> such that the induced character <span class="SimpleMath">\(\chi=\chi(G,K,H)=\theta^G\)</span> is irreducible. Moreover the <em>primitive central idempotent</em> (<a href="chap9_mj.html#X87B6505C7C2EE054"><span class="RefLink">9.4</span></a>) <span class="SimpleMath">\(e_{ℚ }(\chi)\)</span> of <span class="SimpleMath">\(ℚ G\)</span> realized by <span class="SimpleMath">\((K,H)\)</span> is <span class="SimpleMath">\(e(G,K,H)\)</span>, see <a href="chapBib_mj.html#biBORS">[OdRS04]</a>.</p> +<p>Let <span class="SimpleMath">\((K,H)\)</span> be a strong Shoda pair of <span class="SimpleMath">\(G\)</span>. Then <span class="SimpleMath">\((K,H)\)</span> is a Shoda pair (<a href="chap9_mj.html#X7D518BAB80EDE190"><span class="RefLink">9.14</span></a>) of <span class="SimpleMath">\(G\)</span>. Thus there is a linear character <span class="SimpleMath">\(\theta\)</span> of <span class="SimpleMath">\(K\)</span> with kernel <span class="SimpleMath">\(H\)</span> such that the induced character <span class="SimpleMath">\(\chi=\chi(G,K,H)=\theta^G\)</span> is irreducible. Moreover the <em>primitive central idempotent</em> (<a href="chap9_mj.html#X87B6505C7C2EE054"><span class="RefLink">9.4</span></a>) <span class="SimpleMath">\(e_{ℚ }(\chi)\)</span> of <span class="SimpleMath">\(ℚ G\)</span> realized by <span class="SimpleMath">\((K,H)\)</span> is <span class="SimpleMath">\(e(G,K,H)\)</span>, see <a href="chapBib_mj.html#biBORS">[OdS04]</a>.</p> <p>Two <em>strong Shoda pairs</em> (<a href="chap9_mj.html#X7E3479527BAE5B9E"><span class="RefLink">9.15</span></a>) <span class="SimpleMath">\((K_1,H_1)\)</span> and <span class="SimpleMath">\((K_2,H_2)\)</span> of <span class="SimpleMath">\(G\)</span> are said to be <em>equivalent</em> if the characters <span class="SimpleMath">\(\chi(G,K_1,H_1)\)</span> and <span class="SimpleMath">\(\chi(G,K_2,H_2)\)</span> are Galois conjugate, or equivalently if <span class="SimpleMath">\(e(G,K_1,H_1)=e(G,K_2,H_2)\)</span>. A set of representatives of strong Shoda pairs of <span class="SimpleMath">\(G\)</span> is termed as a complete irredundant set of strong Shoda pairs of <span class="SimpleMath">\(G\)</span>.</p> -<p>The advantage of strong Shoda pairs over Shoda pairs is that one can describe the simple algebra <span class="SimpleMath">\(FGe_F(\chi)\)</span> as a matrix algebra of a <em>cyclotomic algebra</em> (<a href="chap9_mj.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>, see <a href="chapBib_mj.html#biBORS">[OdRS04]</a> for <span class="SimpleMath">\(F=ℚ \)</span> and <a href="chapBib_mj.html#biBO">[Olt07]</a> for the general case).</p> +<p>The advantage of strong Shoda pairs over Shoda pairs is that one can describe the simple algebra <span class="SimpleMath">\(FGe_F(\chi)\)</span> as a matrix algebra of a <em>cyclotomic algebra</em> (<a href="chap9_mj.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>, see <a href="chapBib_mj.html#biBORS">[OdS04]</a> for <span class="SimpleMath">\(F=ℚ \)</span> and <a href="chapBib_mj.html#biBO">[Olt07]</a> for the general case).</p> <p>More precisely, <span class="SimpleMath">\(ℚ Ge(G,K,H)\)</span> is isomorphic to <span class="SimpleMath">\(M_n(ℚ (\xi)*_a^t N/K)\)</span>, where <span class="SimpleMath">\(\xi\)</span> is a <span class="SimpleMath">\([K:H]\)</span>-th root of unity, <span class="SimpleMath">\(N\)</span> is the normalizer of <span class="SimpleMath">\(H\)</span> in <span class="SimpleMath">\(G\)</span>, <span class="SimpleMath">\(n=[G:N]\)</span> and <span class="SimpleMath">\(ℚ (\xi)*_a^t N/K\)</span> is a <em>crossed product</em> (see <a href="chap9_mj.html#X7FB21779832CE1CB"><span class="RefLink">9.6</span></a>) with action <span class="SimpleMath">\(a\)</span> and twisting <span class="SimpleMath">\(t\)</span> given as follows:</p> @@ -497,7 +497,7 @@ t(r,s) = \xi^j, \mbox{ if } \varphi(rs)^{-1} \varphi(r)\varphi(s) = x^j, \]</p> -<p>for <span class="SimpleMath">\(r,s \in N/K\)</span> and integers <span class="SimpleMath">\(i\)</span> and <span class="SimpleMath">\(j\)</span>, see <a href="chapBib_mj.html#biBORS">[OdRS04]</a>. Notice that the cocycle is the one given by the natural extension</p> +<p>for <span class="SimpleMath">\(r,s \in N/K\)</span> and integers <span class="SimpleMath">\(i\)</span> and <span class="SimpleMath">\(j\)</span>, see <a href="chapBib_mj.html#biBORS">[OdS04]</a>. Notice that the cocycle is the one given by the natural extension</p> <p class="center">\[ 1 \rightarrow K/H \rightarrow N/H \rightarrow N/K \rightarrow 1 @@ -531,7 +531,7 @@ <p>The group <span class="SimpleMath">\(G\)</span> is <em>strongly monomial</em> if every irreducible character of <span class="SimpleMath">\(G\)</span> is strongly monomial.</p> -<p>Strong Shoda pairs where firstly introduced by Olivieri, del Río and Simón who proved that every abelian-by-supersolvable group is strongly monomial <a href="chapBib_mj.html#biBORS">[OdRS04]</a>. The algorithm to compute the Wedderburn decomposition of rational group algebras for strongly monomial groups was explained in <a href="chapBib_mj.html#biBOR">[OdR03]</a>. This method was extended for semisimple finite group algebras by Broche Cristo and del Río in <a href="chapBib_mj.html#biBBR">[BdR07]</a> (see Section <a href="chap9_mj.html#X800D8C5087D79DC8"><span class="RefLink">9.19</span></a>). Finally, Olteanu <a href="chapBib_mj.html#biBO">[Olt07]</a> shows how to compute the <em>Wedderburn decomposition</em> (<a href="chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of an arbitrary semisimple group ring by making use of not only the strong Shoda pairs of <span class="SimpleMath">\(G\)</span> but also the strong Shoda pairs of the subgroups of <span clas s="SimpleMath">\(G\)</span>.</p> +<p>Strong Shoda pairs where firstly introduced by Olivieri, del Río and Simón who proved that every abelian-by-supersolvable group is strongly monomial <a href="chapBib_mj.html#biBORS">[OdS04]</a>. The algorithm to compute the Wedderburn decomposition of rational group algebras for strongly monomial groups was explained in <a href="chapBib_mj.html#biBOR">[Od03]</a>. This method was extended for semisimple finite group algebras by Broche Cristo and del Río in <a href="chapBib_mj.html#biBBR">[Bd07]</a> (see Section <a href="chap9_mj.html#X800D8C5087D79DC8"><span class="RefLink">9.19</span></a>). Finally, Olteanu <a href="chapBib_mj.html#biBO">[Olt07]</a> shows how to compute the <em>Wedderburn decomposition</em> (<a href="chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of an arbitrary semisimple group ring by making use of not only the strong Shoda pairs of <span class="SimpleMath">\(G\)</span> but also the strong Shoda pairs of the subgroups of <span class=" SimpleMath">\(G\)</span>.</p> <p><a id="X7C8D47C180E0ACAD" name="X7C8D47C180E0ACAD"></a></p> @@ -569,9 +569,9 @@ <p>where <span class="SimpleMath">\(c\)</span> is an arbitrary element of <span class="SimpleMath">\(C\)</span> and <span class="SimpleMath">\(tr\)</span> is the trace map of the field extension <span class="SimpleMath">\(F(\xi)/F\)</span>. Then <span class="SimpleMath">\(\varepsilon_C(K,H)\)</span> does not depend on the choice of <span class="SimpleMath">\(c \in C\)</span> and is a <em>primitive central idempotent</em> (<a href="chap9_mj.html#X87B6505C7C2EE054"><span class="RefLink">9.4</span></a>) of <span class="SimpleMath">\(FK\)</span>.</p> -<p>Finally, let <span class="SimpleMath">\(e_C(G,K,H)\)</span> denote the sum of the different <span class="SimpleMath">\(G\)</span>-conjugates of <span class="SimpleMath">\(\varepsilon_C(K,H)\)</span>. Then <span class="SimpleMath">\(e_C(G,K,H)\)</span> is a <em>primitive central idempotent</em> (<a href="chap9_mj.html#X87B6505C7C2EE054"><span class="RefLink">9.4</span></a>) of <span class="SimpleMath">\(FG\)</span> <a href="chapBib_mj.html#biBBR">[BdR07]</a>. We say that <span class="SimpleMath">\(e_C(G,K,H)\)</span> is the primitive central idempotent realized by the strong Shoda pair <span class="SimpleMath">\((K,H)\)</span> of the group <span class="SimpleMath">\(G\)</span> and the cyclotomic class <span class="SimpleMath">\(C\)</span>.</p> +<p>Finally, let <span class="SimpleMath">\(e_C(G,K,H)\)</span> denote the sum of the different <span class="SimpleMath">\(G\)</span>-conjugates of <span class="SimpleMath">\(\varepsilon_C(K,H)\)</span>. Then <span class="SimpleMath">\(e_C(G,K,H)\)</span> is a <em>primitive central idempotent</em> (<a href="chap9_mj.html#X87B6505C7C2EE054"><span class="RefLink">9.4</span></a>) of <span class="SimpleMath">\(FG\)</span> <a href="chapBib_mj.html#biBBR">[Bd07]</a>. We say that <span class="SimpleMath">\(e_C(G,K,H)\)</span> is the primitive central idempotent realized by the strong Shoda pair <span class="SimpleMath">\((K,H)\)</span> of the group <span class="SimpleMath">\(G\)</span> and the cyclotomic class <span class="SimpleMath">\(C\)</span>.</p> -<p>If <span class="SimpleMath">\(G\)</span> is <em>strongly monomial</em> (<a href="chap9_mj.html#X84C694978557EFE5"><span class="RefLink">9.17</span></a>) then every primitive central idempotent of <span class="SimpleMath">\(FG\)</span> is realizable by some <em>strong Shoda pair</em> (<a href="chap9_mj.html#X7E3479527BAE5B9E"><span class="RefLink">9.15</span></a>) of <span class="SimpleMath">\(G\)</span> and some cyclotomic class <span class="SimpleMath">\(C\)</span> <a href="chapBib_mj.html#biBBR">[BdR07]</a>. As in the zero characteristic case, this explain how to compute the <em>Wedderburn decomposition</em> (<a href="chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of <span class="SimpleMath">\(FG\)</span> for a finite semisimple algebra of a strongly monomial group (see <a href="chapBib_mj.html#biBBR">[BdR07]</a> for details). For non strongly monomial groups the algorithm to compute the Wedderburn decomposition just uses the Brauer characters.</p> +<p>If <span class="SimpleMath">\(G\)</span> is <em>strongly monomial</em> (<a href="chap9_mj.html#X84C694978557EFE5"><span class="RefLink">9.17</span></a>) then every primitive central idempotent of <span class="SimpleMath">\(FG\)</span> is realizable by some <em>strong Shoda pair</em> (<a href="chap9_mj.html#X7E3479527BAE5B9E"><span class="RefLink">9.15</span></a>) of <span class="SimpleMath">\(G\)</span> and some cyclotomic class <span class="SimpleMath">\(C\)</span> <a href="chapBib_mj.html#biBBR">[Bd07]</a>. As in the zero characteristic case, this explain how to compute the <em>Wedderburn decomposition</em> (<a href="chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of <span class="SimpleMath">\(FG\)</span> for a finite semisimple algebra of a strongly monomial group (see <a href="chapBib_mj.html#biBBR">[Bd07]</a> for details). For non strongly monomial groups the algorithm to compute the Wedderburn decomposition just uses the Brauer characters.</p> <p>.</p> @@ -605,7 +605,7 @@ <p>When <span class="SimpleMath">\(R\)</span> is a semisimple ring, then every left ideal <span class="SimpleMath">\(L\)</span> of <span class="SimpleMath">\(R\)</span> is of the form <span class="SimpleMath">\(L=Re\)</span>, where <span class="SimpleMath">\(e\)</span> is an idempotent of <span class="SimpleMath">\(R\)</span>. Therefore, we can use the idempotents to characterize the decompositions of semisimple rings as a direct sum of minimal left ideals. In particular, let <span class="SimpleMath">\(R=\oplus_{i=1}^t L_i\)</span> be a decomposition of a semisimple ring as a direct sum of minimal left ideals. Then, there exists a family <span class="SimpleMath">\(\{e_1,\dots,e_t\}\)</span> of elements of <span class="SimpleMath">\(R\)</span> such that: each <span class="SimpleMath">\(e_i\neq 0\)</span> is an idempotent element, if <span class="SimpleMath">\(i\neq j\)</span>, then <span class="SimpleMath">\(e_ie_j=0\)</span>, <span class="SimpleMath">\(1=e_1+\cdots+e_t\)</span> and each <span class="SimpleMath">\(e_i\)</span> cannot be written as <span class="SimpleMath">\(e_i=e_i'+e_i''\)</span>, where <span class="SimpleMath">\(e_i',e_i''\)</span> are idempotents such that <span class="SimpleMath">\(e_i',e_i''\neq 0\)</span> and <span class="SimpleMath">\(e_i'e_i''=0\)</span>, <span class="SimpleMath">\(1\leq i\leq \)</span>. Conversely, if there exists a family of idempotents <span class="SimpleMath">\(\{e_1,\dots,e_t\}\)</span> satisfying the previous conditions, then the left ideals <span class="SimpleMath">\(L_i=Re_i\)</span> are minimal and <span class="SimpleMath">\(R=\oplus_{i=1}^t L_i\)</span>. Such a set of idempotents is called a <em>complete set of orthogonal primitive idempotents</em> of the ring <span class="SimpleMath">\(R\)</span>. Such a set is not uniquely determined.</p> -<p>Let <span class="SimpleMath">\(\mathbb F\)</span> be a finite field and <span class="SimpleMath">\(G\)</span> a finite nilpotent group such that <span class="SimpleMath">\(\mathbb F G\)</span> is semisimple. Let <span class="SimpleMath">\((H,K)\)</span> be a strong Shoda pair of <span class="SimpleMath">\(G\)</span>, <span class="SimpleMath">\(C\in\mathcal{C}(H/K)\)</span> and set <span class="SimpleMath">\(e_C=e_C(G,H,K)\)</span>, <span class="SimpleMath">\(\varepsilon_C=\varepsilon_C(H,K)\)</span>, <span class="SimpleMath">\(H/K=\langle\overline{a}\rangle\)</span>, <span class="SimpleMath">\(E=E_G(H/K)\)</span>. Let <span class="SimpleMath">\(E_2/K\)</span> and <span class="SimpleMath">\(H_2/K=\langle\overline{a_2}\rangle\)</span> (respectively <span class="SimpleMath">\(E_{2'}/K\)</span> and <span class="SimpleMath">\(H_{2'}/K=\langle\overline{a_{2'}}\rangle\)</span>) denote the 2-parts (respectively 2'-parts) of <span class="SimpleMath">\(E/K\)</span> and <span class="SimpleM ath">\(H/K\)</span> respectively. Then <span class="SimpleMath">\(\langle\overline{a_{2'}}\rangle\)</span> has a cyclic complement <span class="SimpleMath">\(\langle\overline{b_{2'}}\rangle\)</span> in <span class="SimpleMath">\(E_{2'}/K\)</span>. Using the description of the primitive central idempotents and the Wedderburn components of a semisimple finite group algebra <span class="SimpleMath">\(F G\)</span> (<a href="chap9_mj.html#X800D8C5087D79DC8"><span class="RefLink">9.19</span></a>), a complete set of orthogonal primitive idempotents of <span class="SimpleMath">\(\mathbb F Ge_C\)</span> is described (see <a href="chapBib_mj.html#biBOV">[OVG11]</a>) as the set of conjugates of <span class="SimpleMath">\(\beta_{e_C}=\widetilde{b_{2'}}\beta_2\varepsilon_C\)</span> by the elements of <span class="SimpleMath">\(T_{e_C}=T_{2'}T_2T_E\)</span>, where <span class="SimpleMath">\(T_{2'}=\{1,a_{2'},a_{2'}^2,\dots,a_{2'}^{[E_{2'}:H_{2'}]-1}\}\)</span>, <span class="SimpleMath">\(T_E\)</s pan> denotes a right transversal of <span class="SimpleMath">\(E\)</span> in <span class="SimpleMath">\(G\)</span> and <span class="SimpleMath">\(\beta_2\)</span> and <span class="SimpleMath">\(T_2\)</span> are given according to the cases below.</p> +<p>Let <span class="SimpleMath">\(\mathbb F\)</span> be a finite field and <span class="SimpleMath">\(G\)</span> a finite nilpotent group such that <span class="SimpleMath">\(\mathbb F G\)</span> is semisimple. Let <span class="SimpleMath">\((H,K)\)</span> be a strong Shoda pair of <span class="SimpleMath">\(G\)</span>, <span class="SimpleMath">\(C\in\mathcal{C}(H/K)\)</span> and set <span class="SimpleMath">\(e_C=e_C(G,H,K)\)</span>, <span class="SimpleMath">\(\varepsilon_C=\varepsilon_C(H,K)\)</span>, <span class="SimpleMath">\(H/K=\langle\overline{a}\rangle\)</span>, <span class="SimpleMath">\(E=E_G(H/K)\)</span>. Let <span class="SimpleMath">\(E_2/K\)</span> and <span class="SimpleMath">\(H_2/K=\langle\overline{a_2}\rangle\)</span> (respectively <span class="SimpleMath">\(E_{2'}/K\)</span> and <span class="SimpleMath">\(H_{2'}/K=\langle\overline{a_{2'}}\rangle\)</span>) denote the 2-parts (respectively 2'-parts) of <span class="SimpleMath">\(E/K\)</span> and <span class="SimpleM ath">\(H/K\)</span> respectively. Then <span class="SimpleMath">\(\langle\overline{a_{2'}}\rangle\)</span> has a cyclic complement <span class="SimpleMath">\(\langle\overline{b_{2'}}\rangle\)</span> in <span class="SimpleMath">\(E_{2'}/K\)</span>. Using the description of the primitive central idempotents and the Wedderburn components of a semisimple finite group algebra <span class="SimpleMath">\(F G\)</span> (<a href="chap9_mj.html#X800D8C5087D79DC8"><span class="RefLink">9.19</span></a>), a complete set of orthogonal primitive idempotents of <span class="SimpleMath">\(\mathbb F Ge_C\)</span> is described (see <a href="chapBib_mj.html#biBOV">[OV11]</a>) as the set of conjugates of <span class="SimpleMath">\(\beta_{e_C}=\widetilde{b_{2'}}\beta_2\varepsilon_C\)</span> by the elements of <span class="SimpleMath">\(T_{e_C}=T_{2'}T_2T_E\)</span>, where <span class="SimpleMath">\(T_{2'}=\{1,a_{2'},a_{2'}^2,\dots,a_{2'}^{[E_{2'}:H_{2'}]-1}\}\)</span>, <span class="SimpleMath">\(T_E\)</sp an> denotes a right transversal of <span class="SimpleMath">\(E\)</span> in <span class="SimpleMath">\(G\)</span> and <span class="SimpleMath">\(\beta_2\)</span> and <span class="SimpleMath">\(T_2\)</span> are given according to the cases below.</p> <ol> <li><p>If <span class="SimpleMath">\(H_2/K\)</span> has a complement <span class="SimpleMath">\(M_2/K\)</span> in <span class="SimpleMath">\(E_2/K\)</span> then <span class="SimpleMath">\(\beta_2=\widetilde{M_2}\)</span>. Moreover, if <span class="SimpleMath">\(M_2/K\)</span> is cyclic, then there exists <span class="SimpleMath">\(b_2\in E_2\)</span> such that <span class="SimpleMath">\(E_2/K\)</span> is given by the following presentation</p> @@ -674,7 +674,7 @@ \{x\widehat{T_1}\varepsilon x^{-1} \mid x\in T_2\langle{x_e}\rangle\} \]</p> -<p>is a complete set of orthogonal primitive idempotents of <span class="SimpleMath">\(\mathbb F G e\)</span> where <span class="SimpleMath">\(x_e=\psi^{-1}(PAP^{-1})\)</span>, <span class="SimpleMath">\(T_1\)</span> is a transversal of <span class="SimpleMath">\(H\)</span> in <span class="SimpleMath">\(E\)</span> and <span class="SimpleMath">\(T_2\)</span> is a right transversal of <span class="SimpleMath">\(E\)</span> in <span class="SimpleMath">\(G\)</span> (<a href="chapBib_mj.html#biBOV2">[OVG15]</a>). By <span class="SimpleMath">\(\widehat{T_1}\)</span> we denote the element <span class="SimpleMath">\(\frac{1}{|T_1|}\sum_{t\in T_1}{t}\)</span> in <span class="SimpleMath">\(\mathbb F G\)</span>.</p> +<p>is a complete set of orthogonal primitive idempotents of <span class="SimpleMath">\(\mathbb F G e\)</span> where <span class="SimpleMath">\(x_e=\psi^{-1}(PAP^{-1})\)</span>, <span class="SimpleMath">\(T_1\)</span> is a transversal of <span class="SimpleMath">\(H\)</span> in <span class="SimpleMath">\(E\)</span> and <span class="SimpleMath">\(T_2\)</span> is a right transversal of <span class="SimpleMath">\(E\)</span> in <span class="SimpleMath">\(G\)</span> (<a href="chapBib_mj.html#biBOV2">[OV15]</a>). By <span class="SimpleMath">\(\widehat{T_1}\)</span> we denote the element <span class="SimpleMath">\(\frac{1}{|T_1|}\sum_{t\in T_1}{t}\)</span> in <span class="SimpleMath">\(\mathbb F G\)</span>.</p> <p><a id="X856D7975810BF987" name="X856D7975810BF987"></a></p> diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn' '--exclude=.svnignore' old/wedderga-4.11.3/doc/chapBib.html new/wedderga-4.12.0/doc/chapBib.html --- old/wedderga-4.11.3/doc/chapBib.html 2025-12-26 01:15:30.000000000 +0100 +++ new/wedderga-4.12.0/doc/chapBib.html 2026-08-11 02:00:00.000000000 +0200 @@ -27,7 +27,7 @@ <p><a id="biBBR" name="biBBR"></a></p> <p class='BibEntry'> -[<span class='BibKeyLink'><a href="https://www.ams.org/mathscinet-getitem?mr=MR2284667";>BdR07</a></span>] <b class='BibAuthor'>Broche, O. and del Río, Á.</b>, +[<span class='BibKeyLink'><a href="https://www.ams.org/mathscinet-getitem?mr=MR2284667";>Bd07</a></span>] <b class='BibAuthor'>Broche, O. and del Río, Á.</b>, <i class='BibTitle'>Wedderburn decomposition of finite group algebras</i>, <span class='BibJournal'>Finite Fields Appl.</span>, <em class='BibVolume'>13</em> (<span class='BibNumber'>1</span>) @@ -106,7 +106,7 @@ <p><a id="biBOR" name="biBOR"></a></p> <p class='BibEntry'> -[<span class='BibKeyLink'><a href="https://www.ams.org/mathscinet-getitem?mr=MR1981041";>OdR03</a></span>] <b class='BibAuthor'>Olivieri, A. and del Río, Á.</b>, +[<span class='BibKeyLink'><a href="https://www.ams.org/mathscinet-getitem?mr=MR1981041";>Od03</a></span>] <b class='BibAuthor'>Olivieri, A. and del Río, Á.</b>, <i class='BibTitle'>An algorithm to compute the primitive central idempotents and the Wedderburn decomposition of a rational group algebra</i>, <span class='BibJournal'>J. Symbolic Comput.</span>, @@ -118,7 +118,7 @@ <p><a id="biBORS" name="biBORS"></a></p> <p class='BibEntry'> -[<span class='BibKeyLink'><a href="https://www.ams.org/mathscinet-getitem?mr=MR2100373";>OdRS04</a></span>] <b class='BibAuthor'>Olivieri, A., del Río, Á. and Simón, J. J.</b>, +[<span class='BibKeyLink'><a href="https://www.ams.org/mathscinet-getitem?mr=MR2100373";>OdS04</a></span>] <b class='BibAuthor'>Olivieri, A., del Río, Á. and Simón, J. J.</b>, <i class='BibTitle'>On monomial characters and central idempotents of rational group algebras</i>, <span class='BibJournal'>Comm. Algebra</span>, @@ -142,7 +142,7 @@ <p><a id="biBOV" name="biBOV"></a></p> <p class='BibEntry'> -[<span class='BibKey'>OVG11</span>] <b class='BibAuthor'>Olteanu, G. and Van Gelder, I.</b>, +[<span class='BibKey'>OV11</span>] <b class='BibAuthor'>Olteanu, G. and Van Gelder, I.</b>, <i class='BibTitle'>Finite group algebras of nilpotent groups: A complete set of orthogonal primitive idempotents</i>, <span class='BibJournal'>Finite Fields Appl.</span>, <em class='BibVolume'>17</em> (<span class='BibNumber'>2</span>) @@ -153,7 +153,7 @@ <p><a id="biBOV2" name="biBOV2"></a></p> <p class='BibEntry'> -[<span class='BibKey'>OVG15</span>] <b class='BibAuthor'>Olteanu, G. and Van Gelder, I.</b>, +[<span class='BibKey'>OV15</span>] <b class='BibAuthor'>Olteanu, G. and Van Gelder, I.</b>, <i class='BibTitle'>Construction of minimal non-abelian left group codes</i>, <span class='BibJournal'>Des. Codes Cryptography</span>, <em class='BibVolume'>75</em> (<span class='BibNumber'>3</span>) diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn' '--exclude=.svnignore' old/wedderga-4.11.3/doc/chapBib.txt new/wedderga-4.12.0/doc/chapBib.txt --- old/wedderga-4.11.3/doc/chapBib.txt 2025-12-26 01:15:30.000000000 +0100 +++ new/wedderga-4.12.0/doc/chapBib.txt 2026-08-11 02:00:00.000000000 +0200 @@ -2,7 +2,7 @@ [1XReferences[101X - [[20XBdR07[120X] [16XBroche, O. and del Río, Á.[116X, [17XWedderburn decomposition of finite group + [[20XBd07[120X] [16XBroche, O. and del Río, Á.[116X, [17XWedderburn decomposition of finite group algebras[117X, [18XFinite Fields Appl.[118X, [19X13[119X, 1 (2007), 71–79. [[20XBen76[120X] [16XBenard, M.[116X, [17XSchur indices and cyclic defect groups[117X, [18XAnn. of Math. @@ -24,11 +24,11 @@ Mathematical Society, Lecture Note Series, [19X250[119X, Cambridge, UK (1998), x+287 pages. - [[20XOdR03[120X] [16XOlivieri, A. and del Río, Á.[116X, [17XAn algorithm to compute the primitive + [[20XOd03[120X] [16XOlivieri, A. and del Río, Á.[116X, [17XAn algorithm to compute the primitive central idempotents and the Wedderburn decomposition of a rational group algebra[117X, [18XJ. Symbolic Comput.[118X, [19X35[119X, 6 (2003), 673–687. - [[20XOdRS04[120X] [16XOlivieri, A., del Río, Á. and Simón, J. J.[116X, [17XOn monomial characters + [[20XOdS04[120X] [16XOlivieri, A., del Río, Á. and Simón, J. J.[116X, [17XOn monomial characters and central idempotents of rational group algebras[117X, [18XComm. Algebra[118X, [19X32[119X, 4 (2004), 1531–1550. @@ -36,11 +36,11 @@ algebras by the Brauer-Witt theorem[117X, [18XMath. Comp.[118X, [19X76[119X, 258 (2007), 1073–1087 (electronic). - [[20XOVG11[120X] [16XOlteanu, G. and Van Gelder, I.[116X, [17XFinite group algebras of nilpotent + [[20XOV11[120X] [16XOlteanu, G. and Van Gelder, I.[116X, [17XFinite group algebras of nilpotent groups: A complete set of orthogonal primitive idempotents[117X, [18XFinite Fields Appl.[118X, [19X17[119X, 2 (2011), 157–165. - [[20XOVG15[120X] [16XOlteanu, G. and Van Gelder, I.[116X, [17XConstruction of minimal non-abelian + [[20XOV15[120X] [16XOlteanu, G. and Van Gelder, I.[116X, [17XConstruction of minimal non-abelian left group codes[117X, [18XDes. Codes Cryptography[118X, [19X75[119X, 3 (2015), 359–373. [[20XPas89[120X] [16XPassman, D. S.[116X, [17XInfinite crossed products[117X, Academic Press Inc., Pure diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn' '--exclude=.svnignore' old/wedderga-4.11.3/doc/chapBib_mj.html new/wedderga-4.12.0/doc/chapBib_mj.html --- old/wedderga-4.11.3/doc/chapBib_mj.html 2025-12-26 01:15:30.000000000 +0100 +++ new/wedderga-4.12.0/doc/chapBib_mj.html 2026-08-11 02:00:00.000000000 +0200 @@ -30,7 +30,7 @@ <p><a id="biBBR" name="biBBR"></a></p> <p class='BibEntry'> -[<span class='BibKeyLink'><a href="https://www.ams.org/mathscinet-getitem?mr=MR2284667";>BdR07</a></span>] <b class='BibAuthor'>Broche, O. and del Río, Á.</b>, +[<span class='BibKeyLink'><a href="https://www.ams.org/mathscinet-getitem?mr=MR2284667";>Bd07</a></span>] <b class='BibAuthor'>Broche, O. and del Río, Á.</b>, <i class='BibTitle'>Wedderburn decomposition of finite group algebras</i>, <span class='BibJournal'>Finite Fields Appl.</span>, <em class='BibVolume'>13</em> (<span class='BibNumber'>1</span>) @@ -109,7 +109,7 @@ <p><a id="biBOR" name="biBOR"></a></p> <p class='BibEntry'> -[<span class='BibKeyLink'><a href="https://www.ams.org/mathscinet-getitem?mr=MR1981041";>OdR03</a></span>] <b class='BibAuthor'>Olivieri, A. and del Río, Á.</b>, +[<span class='BibKeyLink'><a href="https://www.ams.org/mathscinet-getitem?mr=MR1981041";>Od03</a></span>] <b class='BibAuthor'>Olivieri, A. and del Río, Á.</b>, <i class='BibTitle'>An algorithm to compute the primitive central idempotents and the Wedderburn decomposition of a rational group algebra</i>, <span class='BibJournal'>J. Symbolic Comput.</span>, @@ -121,7 +121,7 @@ <p><a id="biBORS" name="biBORS"></a></p> <p class='BibEntry'> -[<span class='BibKeyLink'><a href="https://www.ams.org/mathscinet-getitem?mr=MR2100373";>OdRS04</a></span>] <b class='BibAuthor'>Olivieri, A., del Río, Á. and Simón, J. J.</b>, +[<span class='BibKeyLink'><a href="https://www.ams.org/mathscinet-getitem?mr=MR2100373";>OdS04</a></span>] <b class='BibAuthor'>Olivieri, A., del Río, Á. and Simón, J. J.</b>, <i class='BibTitle'>On monomial characters and central idempotents of rational group algebras</i>, <span class='BibJournal'>Comm. Algebra</span>, @@ -145,7 +145,7 @@ <p><a id="biBOV" name="biBOV"></a></p> <p class='BibEntry'> -[<span class='BibKey'>OVG11</span>] <b class='BibAuthor'>Olteanu, G. and Van Gelder, I.</b>, +[<span class='BibKey'>OV11</span>] <b class='BibAuthor'>Olteanu, G. and Van Gelder, I.</b>, <i class='BibTitle'>Finite group algebras of nilpotent groups: A complete set of orthogonal primitive idempotents</i>, <span class='BibJournal'>Finite Fields Appl.</span>, <em class='BibVolume'>17</em> (<span class='BibNumber'>2</span>) @@ -156,7 +156,7 @@ <p><a id="biBOV2" name="biBOV2"></a></p> <p class='BibEntry'> -[<span class='BibKey'>OVG15</span>] <b class='BibAuthor'>Olteanu, G. and Van Gelder, I.</b>, +[<span class='BibKey'>OV15</span>] <b class='BibAuthor'>Olteanu, G. and Van Gelder, I.</b>, <i class='BibTitle'>Construction of minimal non-abelian left group codes</i>, <span class='BibJournal'>Des. Codes Cryptography</span>, <em class='BibVolume'>75</em> (<span class='BibNumber'>3</span>) diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn' '--exclude=.svnignore' old/wedderga-4.11.3/doc/chooser.html new/wedderga-4.12.0/doc/chooser.html --- old/wedderga-4.11.3/doc/chooser.html 2025-12-26 01:15:30.000000000 +0100 +++ new/wedderga-4.12.0/doc/chooser.html 2026-08-11 02:00:00.000000000 +0200 @@ -23,9 +23,18 @@ return ""; } +<!-- load dark appearance either explicitly or via the OS preference --> +function writeAppearanceStyle(mode) { + if (mode == "dark") { + document.writeln('<link rel="stylesheet" type="text/css" href="dark.css" />'); + } else if (mode != "light") { + document.writeln('<style type="text/css">@import "dark.css" (prefers-color-scheme: dark);</style>'); + } +} + <!-- find style from current values in form --> function getstyle() { - var choices = ["toggle","colorprompt","tocside","font","justify"]; + var choices = ["toggle","colorprompt","tocside","font","justify","appearance"]; var style = ""; for (var i=0; choices.length > i; i++) { var a = currval(choices[i]); @@ -55,17 +64,23 @@ <!-- initialize form from GAPDocStyle cookie --> function initform() { var style = valueString(document.cookie, "GAPDocStyle"); + if (style == 0 || style == "default") + writeAppearanceStyle(""); if (style != 0 && style.length > 0 && style.length != "default") { stlist = style.split(","); var chform = document.forms[0].elements; + var appearance = ""; for (var i=0; chform.length > i; i++) { if (chform[i].type == "radio") { for (var j=0; stlist.length > j; j++) { - if (chform[i].value == stlist[j]) + if (stlist[j] == "dark" || stlist[j] == "light") + appearance = stlist[j]; + if (chform[i].value == stlist[j]) chform[i].checked = true; } } } + writeAppearanceStyle(appearance); } } @@ -119,6 +134,15 @@ <input type="radio" name="justify" value="ragged" onclick="javascript:f()"/> ragged right </p> +<p> +Appearance: +<input type="radio" name="appearance" value="" +onclick="javascript:f()"/> system (default) +<input type="radio" name="appearance" value="light" checked="checked" +onclick="javascript:f()"/> light +<input type="radio" name="appearance" value="dark" +onclick="javascript:f()"/> dark +</p> </form> <p> <a name="backLINK" href=""><strong>Apply settings to last page.</strong></a> diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn' '--exclude=.svnignore' old/wedderga-4.11.3/doc/dark.css new/wedderga-4.12.0/doc/dark.css --- old/wedderga-4.11.3/doc/dark.css 1970-01-01 01:00:00.000000000 +0100 +++ new/wedderga-4.12.0/doc/dark.css 2026-08-11 02:00:00.000000000 +0200 @@ -0,0 +1,178 @@ +/* dark.css Frank Luebeck */ +/* Initial dark theme contributed by kiryph in issue #75. */ + +/* colors */ +body { + background: #121212; + color: #eee; +} + +a:link { + color: #576cad; +} + +a:visited { + color: #576cad; +} + +a:active { + color: #eee; +} + +a:hover { + background: #eee; +} + +pre { + color: black; +} + +tt, code { + color: #eee; +} + +/* layout for the definitions of functions, variables, ... */ +div.func { + background: #909090; +} + +/* Example elements (for old converted manuals, now in div+pre */ +table.example { + background: #efefef; +} + +/* becomes ... */ +div.example { + background: #efefef; + color: black; +} + +/* Links to chapters in all files at top and bottom. */ +div.chlinktop { + background: #22272e; + border-color: #3a414a; + color: #d7dde5; +} + +div.chlinktop a:hover { + background: #2f3742; +} + +div.chlinkbot { + background: #22272e; + border-color: #3a414a; + color: #d7dde5; +} + +/* and this is for the "Top", "Prev", "Next" links */ +div.chlinkprevnexttop { + background: #22272e; + border-color: #3a414a; + color: #d7dde5; +} + +div.chlinkprevnexttop a:hover { + background: #2f3742; +} + +div.chlinkprevnextbot { + background: #22272e; + border-color: #3a414a; + color: #d7dde5; +} + +div.chlinkprevnextbot a:hover { + background: #2f3742; +} + +div.ContChap div.ContSect:hover div.ContSSBlock { + background: #eee; + border-color: #666; + color: #000; +} + +div.ContSSBlock a:hover { + background: #fff; +} + +/* and here for the side menu of contents in the chapter files */ +div.ChapSects a:hover { + background: #eee; + color: #000; +} + +div.ChapSects div.ContSect:hover div.ContSSBlock { + background: #b5b5b5; + border-color: #666; + color: #000; +} + +div.ChapSects div.ContSect:hover div.ContSSBlock a:hover { + background: #828282; +} + +/* Table elements */ +table.GAPDocTable { + border-color: black; +} + +table.GAPDocTable td, table.GAPDocTable th { + border-color: #555; +} + +table.GAPDocTablenoborder td, table.GAPDocTable th { + border-color: #555; +} + +/* Colors and fonts can be overwritten for some types of elements. */ +/* Verb elements */ +pre.normal { + color: #eee; +} + +/* Func-like elements and Ref to Func-like */ +code.func { + color: #eee; +} + +/* K elements */ +code.keyw { + color: #983d3d; +} + +/* F elements */ +code.file { + color: #8e4510; +} + +/* Arg elements */ +var.Arg { + color: #060; +} + +/* colors for ColorPrompt like examples */ +span.GAPprompt { + color: #000097; +} + +span.GAPbrkprompt { + color: #970000; +} + +span.GAPinput { + color: #970000; +} + +/* Bib entries */ +span.BibKey { + color: #052; +} + +/* for light and dark mode pictures */ +.only-on-dark { + display: block; +} + +.only-on-light { + display: none; +} diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn' '--exclude=.svnignore' old/wedderga-4.11.3/doc/manual.css new/wedderga-4.12.0/doc/manual.css --- old/wedderga-4.11.3/doc/manual.css 2025-12-26 01:15:30.000000000 +0100 +++ new/wedderga-4.12.0/doc/manual.css 2026-08-11 02:00:00.000000000 +0200 @@ -477,6 +477,10 @@ span.BibHowpublished { } +/* for light and dark mode pictures */ +.only-on-dark { + display: none; +} diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn' '--exclude=.svnignore' old/wedderga-4.11.3/doc/manual.js new/wedderga-4.12.0/doc/manual.js --- old/wedderga-4.11.3/doc/manual.js 2025-12-26 01:15:30.000000000 +0100 +++ new/wedderga-4.12.0/doc/manual.js 2026-08-11 02:00:00.000000000 +0200 @@ -47,6 +47,20 @@ return 0; } +/* load dark appearance either explicitly or via the OS preference */ +function writeAppearanceStyle(mode) { + if (mode == "dark") { + document.writeln( + '<link rel="stylesheet" type="text/css" href="dark.css" />' + ); + } else if (mode != "light") { + document.writeln( + '<style type="text/css">@import "dark.css" ' + + '(prefers-color-scheme: dark);</style>' + ); + } +} + /* when a non-default style is chosen via URL or a cookie, then the cookie is reset and the styles .js and .css files are read */ function overwriteStyle() { @@ -55,6 +69,8 @@ /* otherwise check cookie */ if (style == 0) style = valueString(document.cookie, "GAPDocStyle"); + if (style == 0 || style == "default") + writeAppearanceStyle(""); if (style == 0) return; if (style == "default") @@ -70,13 +86,19 @@ document.cookie = "GAPDocStyle="+style+";Path="+path; /* split into names of style files */ var stlist = style.split(","); + var appearance = ""; /* read style's css and js files */ for (var i=0; i < stlist.length; i++) { - document.writeln('<link rel="stylesheet" type="text/css" href="'+ - stlist[i]+'.css" />'); - document.writeln('<script src="'+stlist[i]+ - '.js" type="text/javascript"></script>'); + if (stlist[i] == "dark" || stlist[i] == "light") { + appearance = stlist[i]; + } else { + document.writeln('<link rel="stylesheet" type="text/css" href="'+ + stlist[i]+'.css" />'); + document.writeln('<script src="'+stlist[i]+ + '.js" type="text/javascript"></script>'); + } } + writeAppearanceStyle(appearance); } } @@ -110,4 +132,3 @@ for (var i=0; i < jscontentfuncs.length; i++) jscontentfuncs[i](); } - Binary files old/wedderga-4.11.3/doc/manual.pdf and new/wedderga-4.12.0/doc/manual.pdf differ diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn' '--exclude=.svnignore' old/wedderga-4.11.3/doc/manual.tex new/wedderga-4.12.0/doc/manual.tex --- old/wedderga-4.11.3/doc/manual.tex 2025-12-26 01:15:30.000000000 +0100 +++ new/wedderga-4.12.0/doc/manual.tex 2026-08-11 02:00:00.000000000 +0200 @@ -90,27 +90,29 @@ \begin{titlepage} \mbox{}\vfill -\begin{center}{\maintitlesize \textbf{\textsf{Wedderga}\mbox{}}}\\ +\begin{center}{\maintitlesize \textbf{\textsf{Wedderga}\\ +\mbox{}}}\\ \vfill -\hypersetup{pdftitle=\textsf{Wedderga}} +\hypersetup{pdftitle={\textsf{Wedderga}}} \markright{\scriptsize \mbox{}\hfill \textsf{Wedderga} \hfill\mbox{}} -{\Huge \textbf{Wedderburn Decomposition of Group Algebras\mbox{}}}\\ +{\Huge \textbf{Wedderburn Decomposition of Group Algebras\\ +\mbox{}}}\\ \vfill -{\Huge Version 4.11.3\mbox{}}\\[1cm] -{26 December 2025\mbox{}}\\[1cm] +{\Huge Version 4.12.0\mbox{}}\\[1cm] +{11 August 2026\mbox{}}\\[1cm] \mbox{}\\[2cm] -{\Large \textbf{Gurmeet Kaur Bakshi \mbox{}}}\\ -{\Large \textbf{Osnel Broche Cristo \mbox{}}}\\ -{\Large \textbf{Allen Herman \mbox{}}}\\ -{\Large \textbf{Olexandr Konovalov \mbox{}}}\\ -{\Large \textbf{Sugandha Maheshwary \mbox{}}}\\ -{\Large \textbf{Aurora Olivieri \mbox{}}}\\ -{\Large \textbf{Gabriela Olteanu \mbox{}}}\\ -{\Large \textbf{{\a'A}ngel del R{\a'\i}o \mbox{}}}\\ -{\Large \textbf{Inneke Van Gelder \mbox{}}}\\ -\hypersetup{pdfauthor=Gurmeet Kaur Bakshi ; Osnel Broche Cristo ; Allen Herman ; Olexandr Konovalov ; Sugandha Maheshwary ; Aurora Olivieri ; Gabriela Olteanu ; {\a'A}ngel del R{\a'\i}o ; Inneke Van Gelder } +{\Large \textbf{\strut Gurmeet Kaur Bakshi \strut\mbox{}}}\\ +{\Large \textbf{\strut Osnel Broche Cristo \strut\mbox{}}}\\ +{\Large \textbf{\strut Allen Herman \strut\mbox{}}}\\ +{\Large \textbf{\strut Olexandr Konovalov \strut\mbox{}}}\\ +{\Large \textbf{\strut Sugandha Maheshwary \strut\mbox{}}}\\ +{\Large \textbf{\strut Aurora Olivieri \strut\mbox{}}}\\ +{\Large \textbf{\strut Gabriela Olteanu \strut\mbox{}}}\\ +{\Large \textbf{\strut {\a'A}ngel del R{\a'\i}o \strut\mbox{}}}\\ +{\Large \textbf{\strut Inneke Van Gelder \strut\mbox{}}}\\ +\hypersetup{pdfauthor={Gurmeet Kaur Bakshi ; Osnel Broche Cristo ; Allen Herman ; Olexandr Konovalov ; Sugandha Maheshwary ; Aurora Olivieri ; Gabriela Olteanu ; {\a'A}ngel del R{\a'\i}o ; Inneke Van Gelder }} \end{center}\vfill \mbox{}\\ @@ -139,7 +141,7 @@ }\\ {\mbox{}\\ \small \noindent \textbf{Olexandr Konovalov } Email: \href{mailto://[email protected]} {\texttt{obk1@st\texttt{\symbol{45}}andrews.ac.uk}}\\ - Homepage: \href{https://alex-konovalov.github.io/} {\texttt{https://alex\texttt{\symbol{45}}konovalov.github.io/}}\\ + Homepage: \href{https://olexandr-konovalov.github.io/} {\texttt{https://olexandr\texttt{\symbol{45}}konovalov.github.io/}}\\ Address: \begin{minipage}[t]{8cm}\noindent School of Computer Science, University of St Andrews\\ Jack Cole Building, North Haugh,\\ @@ -202,7 +204,7 @@ {\small \section*{Copyright} \logpage{[ 0, 0, 2 ]} - {\copyright} 2006\texttt{\symbol{45}}2025 by Gurmeet Kaur Bakshi, Osnel Broche + {\copyright} 2006\texttt{\symbol{45}}2026 by Gurmeet Kaur Bakshi, Osnel Broche Cristo, Allen Herman, Olexandr Konovalov, Sugandha Maheshwary, Aurora Olivieri, Gabriela Olteanu, {\a'A}ngel del R{\a'\i}o and Inneke Van Gelder. @@ -216,7 +218,7 @@ G. K. Bakshi, O. Broche Cristo, A. Herman, O. Konovalov, S. Maheshwary, A. Olivieri, G. Olteanu, {\a'A}. del R{\a'\i}o and I. Van Gelder. \emph{Wedderga \texttt{\symbol{45}}\texttt{\symbol{45}}\texttt{\symbol{45}} -Wedderburn Decomposition of Group Algebras, Version 4.11.3;} 2025 (\href{https://gap-packages.github.io/wedderga/} {\texttt{https://gap\texttt{\symbol{45}}packages.github.io/wedderga/}}). \mbox{}}\\[1cm] +Wedderburn Decomposition of Group Algebras, Version 4.12.0;} 2026 (\href{https://gap-packages.github.io/wedderga/} {\texttt{https://gap\texttt{\symbol{45}}packages.github.io/wedderga/}}). \mbox{}}\\[1cm] {\small \section*{Acknowledgements} \logpage{[ 0, 0, 3 ]} diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn' '--exclude=.svnignore' old/wedderga-4.11.3/doc/manual.xml new/wedderga-4.12.0/doc/manual.xml --- old/wedderga-4.11.3/doc/manual.xml 2025-12-26 01:15:30.000000000 +0100 +++ new/wedderga-4.12.0/doc/manual.xml 2026-08-11 02:00:00.000000000 +0200 @@ -46,7 +46,7 @@ <Author>Olexandr Konovalov <Email>[email protected]</Email> - <Homepage>https://alex-konovalov.github.io/</Homepage> + <Homepage>https://olexandr-konovalov.github.io/</Homepage> <Address> School of Computer Science, University of St Andrews<Br/> Jack Cole Building, North Haugh,<Br/> diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn' '--exclude=.svnignore' old/wedderga-4.11.3/doc/manualbib.xml.bib new/wedderga-4.12.0/doc/manualbib.xml.bib --- old/wedderga-4.11.3/doc/manualbib.xml.bib 2025-12-26 01:15:30.000000000 +0100 +++ new/wedderga-4.12.0/doc/manualbib.xml.bib 2026-08-11 02:00:00.000000000 +0200 @@ -33,7 +33,7 @@ printedkey = {BS72} } @article{ BR, - author = {Broche, O. and del R{\a'\i}o, {\a'A}.}, + author = {Broche, O. and {del R{\a'\i}o}, {\a'A}.}, title = {Wedderburn decomposition of finite group algebras}, journal = {Finite Fields Appl.}, volume = {13}, @@ -110,7 +110,7 @@ printedkey = {BM16} } @article{ OR, - author = {Olivieri, A. and del R{\a'\i}o, {\a'A}.}, + author = {Olivieri, A. and {del R{\a'\i}o}, {\a'A}.}, title = {An algorithm to compute the primitive central idempotents and the {W}edderburn decomposition of a rational group algebra}, @@ -127,7 +127,7 @@ printedkey = {OR03} } @article{ ORS, - author = {Olivieri, A. and del R{\a'\i}o, {\a'A}. and + author = {Olivieri, A. and {del R{\a'\i}o}, {\a'A}. and Sim{\a'o}n, J. J.}, title = {On monomial characters and central idempotents of rational group algebras}, @@ -163,7 +163,7 @@ printedkey = {Olt07} } @article{ OV, - author = {Olteanu, G. and Van Gelder, I.}, + author = {Olteanu, G. and {Van Gelder}, I.}, title = {Finite group algebras of nilpotent groups: A complete set of orthogonal primitive idempotents}, journal = {Finite Fields Appl.}, @@ -174,7 +174,7 @@ printedkey = {OG11} } @article{ OV2, - author = {Olteanu, G. and Van Gelder, I.}, + author = {Olteanu, G. and {Van Gelder}, I.}, title = {Construction of minimal non\texttt{\symbol{45}}abelian left group codes}, journal = {Des. Codes Cryptography},
