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Idempotenten in Groepringen

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Idempotenten in Groepringen FACULTEIT WETENSCHAPPEN EN BIO-INGENIEURSWETENSCHAPPEN DEPARTEMENT WISKUNDE Idempotenten in Groepringen Proefschrift ingediend met het oog op het behalen van de graad van Master in de Wiskunde Inneke Van Gelder Promotor: Prof. Dr. Eric Jespers MEI 2010 FACULTY OF SCIENCE AND BIO-ENGINEERING SCIENCES DEPARTMENT OF MATHEMATICS Idempotents in Group Rings Graduation thesis submitted in partial fulfillment of the requirements for the degree of Master in Mathematics Inneke Van Gelder Promotor: Prof. Dr. Eric Jespers MAY 2010 Voorwoord De lente van 2005, nu exact vijf jaar geleden, een studiekeuze maken bleek plots een niet al te makkelijke opgave. Hoewel een vijf jaar durende studie eerst vreselijk lang leek, weet ik nu dat ik destijds de juiste keuze heb gemaakt. De afgelopen vijf jaren zijn immers voorbij gevlogen en een diploma komt nu duidelijk in zicht. 'Afstuderen' is vaak een synoniem voor 'thesis schrijven': D´eThesis. Een zware taak die ik vol spanning, maar toch ook met de nodige vrees, tegemoet ging. Die vrees was echter nergens voor nodig. Ik houd er een zeer leuke en leerrijke ervaring aan over. Alleen daarom al kan ik voor mezelf stellen dat ik tevreden ben met het resultaat. Nooit had ik gedacht zoveel te kunnen bereiken op enkele maanden tijd. Graag wil ik in het bijzonder mijn promotor bedanken. Eric Jespers, bedankt voor de goede begeleiding, motivatie en de vrijheid die je me gaf. Tevens bedankt dat je me de kans gaf gedurende een volledige week een conferentie bij te wonen. I would like to thank Angel´ del R´ıo.Thank you for your patient explanations, for your interesting lectures and talks and for the everlasting enthousiasm. And also to Gabriela Olteanu, for our interesting talks involving GAP en for your support. Tenslotte bedank ik ook mijn medestudenten voor de fantastische tijd die we gedurende vijf interessante en onvergetelijke jaren samen hebben beleefd. Inneke Van Gelder Diest - mei 2010 i Inhoudsopgave Inleiding 1 Samenvatting5 1 Preliminaries 14 1.1 Semisimple rings.................................. 14 1.2 Group rings..................................... 16 1.3 Representations and characters.......................... 19 1.4 Crossed products.................................. 22 1.5 Quaternion algebras................................ 24 2 Primitive central idempotents of rational group algebras 25 2.1 Some definitions and background......................... 25 2.2 The primitive central idempotent associated to a monomial character..... 30 2.3 Abelian-by-supersolvable groups.......................... 36 2.4 Example: metacyclic groups............................ 41 2.5 An algorithm to compute the Wedderburn decomposition............ 42 2.6 More examples................................... 49 2.7 The primitive central idempotent associated to an irreducible character.... 61 3 Primitive central idempotents of finite group algebras 65 3.1 Some notations and background.......................... 65 3.2 The primitive central idempotents......................... 66 3.3 Examples...................................... 72 4 A complete set of orthogonal primitive idempotents 74 4.1 The rational group ring of nilpotent groups.................... 74 4.2 Examples...................................... 86 4.3 Finite group rings of nilpotent groups....................... 93 5 Appendix: some GAP functions 98 References 106 Index 110 ii Inleiding De groepentheorie kent een lange geschiedenis1 die teruggaat tot de 17de eeuw, met wortels in het oplossen van algebra¨ısche vergelijkingen en getaltheorie. Toen was er echter nog geen sprake van het concept groep. Het artikel, R´eflections sur la r´esolutionalg´ebriquedes ´equations, uit 1770 van Joseph- Louis Lagrange, gevolgd door artikels van Paolo Ruffini en Niels Henrik Abel, trok de aandacht van vele wiskundigen naar het concept permutaties. In 1830, was Evariste´ Galois de eerste om groepen en deelgroepen van permutaties te gebruiken. Hij hanteerde daarvoor ook de term groep, zoals in de monderne betekenis, maar dan beperkt tot permutaties. Hij introduceerde ook concepten zoals normale deelgroep en oplosbare groep. Augustin Louis Cauchy begreep de relevantie van permutatiegroepen als onafhankelijk onderwerp en schreef er dan ook een reeks interessante artikels over in de periode 1844-1846. Be¨ınvloed door Cauchy, realiseerde Arthur Cayley zich dat de notie van een groep gefor- muleerd kan worden in een meer abstracte context en hij gaf de eerste definitie van een abstracte groep in 1854 [Cay54]. Dit artikel wordt door velen beschouwd als het beginpunt van de abstracte groepentheorie. Uiteindelijk werd de definitie van een abstracte groep zoals we die vandaag gebruiken, gegeven door Walther Franz Anton von Dyck, een student van Felix Christian Klein, in 1882 [vD82]. In 1897 schreef William Burnside het eerste boek volledig gewijd aan groepentheorie, na- melijk Theory of Groups of Finite Order. Dit werk gold tientallen jaren als het standaardwerk over eindige groepen. De ringtheorie is iets recenter dan de groepentheorie. In 1837 publiceerde William Rowan Hamilton ´e´envan zijn belangrijkste artikels over de complexe getallen. De complexe getallen a + bi werden eerder al door Euler en Gauss bestudeerd, maar Hamilton was de eerste die ze expliciet definieerde als een geordend paar (a; b) uit het re¨elevlak. Later zocht hij naar manieren om analoog een driedimensionale ruimte te cre¨eren.Hij slaagde er echter niet in om een goede vermenigvuldiging te defini¨eren.Later werd zelfs getoond dat dit onmogelijk is. Uiteindelijk probeerde hij het met vier dimensies en cre¨erdede quaternionen. Dat zijn elementen van de vorm a + bi + cj + dk, met a; b; c; d re¨elegetallen en i; j; k formele basissymbolen. Ze worden componentsgewijs opgeteld en de vermenigvuldiging verloopt volgens de regel i2 = j2 = k2 = ijk = −1: E´envan de bekenste anekdotes in de geschiedenis van de wiskunde is die van Hamilton die op 16 oktober 1843 samen met zijn vrouw over de Brougham Bridge in Dublin wandelt. Terwijl 1Deze korte geschiedenis komt uit [MS02]. 1 Inleiding zijn vrouw tegen hem praat, luistert hij eigenlijk niet, maar denkt na over de vermenigvuldi- ging van viertallen re¨elegetallen en ontdekt de quaternionen, de eerste niet-commutatieve ring ooit bestudeerd. Uit vrees om de ontdekking te vergeten, kerft hij de vermenigvuldigingsregel in een steen van de brug. Op die plaats is nu nog steeds een gedenkplaat met de welbekende formule te vinden. Datzelfde jaar introduceerde John Thomas Graves, ge¨ınspireerddoor zijn vriend Hamilton, de octonionen, een niet-associatieve uitbreiding van de quaternionen. Ze worden gedefinieerd als elementen van de vorm a0 + a1e1 + a2e2 + ··· + a7e7, met re¨eleco¨effici¨enten ai en ba- siselementen ei. De optelling verloopt weer componentsgewijs en de vermenigvuldiging is gedefinieerd met enkele regels op de basiselementen. Het was echter Arthur Cayley die er in 1845 een eerste artikel over publiceerde, nadat hij de octonionen onafhankelijk van Graves ontdekte. Daarom zijn de octonionen ook gekend als de Cayley getallen. Hamilton breidde de constructie echter nog verder uit en definieerde de biquaternionen. Dat zijn opnieuw elementen van de vorm a + bi + cj + dk, maar met complexe co¨effici¨enten a; b; c; d. Kort daarna introduceerde hij de hypercomplexe systemen. Dat zijn dan weer verza- melingen van elementen van de vorm a1e1 + a2e2 + ··· + anen, waar de optelling components- gewijs verloopt en de vermenigvuldiging gedefinieerd is op de basiselementen. Deze gebeurtenissen kunnen als de eerste stappen in de ontwikkeling van de ringtheorie gezien worden. Vanaf nu zou er meer interesse komen in de structuur van algebra's. In 1870 gaf Benjamin Pierce een classificatie van alle gekende algebra's van dimensie hoogstens 6. Hiertoe introduceerde Pierce enkele zeer belangrijke idee¨enin ringtheorie, zoals de noties van idempotenten en het gebruik van idempotenten om een decompositie van een algebra te bekomen. In die eeuw vonden er ook belangrijke ontwikkelingen plaats in de theorie van niet- associatieve algebra's. Volgend op het werk van Sophus Lie en Wilhelm Karl Joseph Killing in de studie van Lie-groepen en Lie-algebra's, introduceerden Georg Scheffers en Eduard Study eind 19de eeuw enkele basisnoties voor de ontwikkeling in de theorie over de structuur van algebra's, zoals de concepten van simpele en semisimpele algebra's, hoewel ze een andere terminologie gebruikten. Deze resultaten inspireerden zowel Theodor Molien als Elie´ Cartan. Zij verkregen, on- afhankelijk van elkaar, belangrijke resultaten over de structuur van eindigdimensionale re¨ele en complexe algebra's. Hiervoor introduceerden ze in deze context de noties van simpele en semisimpele algebra's en karakteriseerden ze simpele algebra's als volledige matrixalgebra's. Dit werk bereikte zijn hoogtepunt in de mooie stellingen van Joseph Henry Maclagan Wedderburn, die de structuur van eindigdimensionale algebra's over willekeurige lichamen beschrijven. Hiertoe gebruikte hij technieken gerelateerd met het bestaan van idempotente elementen, zoals gesuggereerd in het eerder werk van Pierce. Zoals eerder vermeld, gaf Cayley de eerste definitie van een abstracte groep. Het is inte- ressant om te vermelden dat in datzelfde artikel ook de notie van groepring verscheen voor de eerste keer. Hoewel hij precies dezelfde definitie gebruikte zoals we die de dag van vandaag hanteren, kende zijn werk weinig invloed en groepringen bleven nog een tijdje onbekend. Groepringen werden opnieuw ingevoerd door Molien wanneer hij zich realiseerde dat dit
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