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Idempotents in Group Algebras and Applications to Units

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Idempotents in Group Algebras and Applications to Units Vrije Universiteit Brussel - Faculty of Science and Bio-engineering Sciences Group representations: idempotents in group algebras and applications to units Graduation thesis submitted in fulfillment of the requirements for the degree of Doctor in Sciences Author: Promotor: Inneke Van Gelder Prof. Dr. Eric Jespers Copromotor: Prof. Dr. Gabriela Olteanu march 2015 ACKNOWLEDGMENTS Ik zou graag iedereen willen bedanken die, elk op zijn manier, heeft bijgedragen tot de realisatie van deze doctoraatsthesis. Eerst en vooral zou ik graag mijn promotor Eric Jespers willen bedanken. Hij heeft zijn enthousiasme voor groepsringen op mij overgedragen door zijn interessante cursussen en boeiende lezingen. Door zijn ruime kennissenkring kwam ik in contact met verschillende vriendelijke wiskundigen waar ik veel van heb kunnen leren. I owe a very special thank you to my copromotor Gabriela Olteanu. I would like to thank her for her hospitality during my stays in Romania, for her support, both professional and personal. Thanks for the nice cooperations and the very careful proofreading of our joint works. Also my gratitude is expressed to my co-authors Andreas B¨achle, Mauricio Caicedo, Angel´ del R´ıo,Florian Eisele and Ann Kiefer for the pleasant and enriching collaborations. Thanks to Allen Herman and Alexander Konovalov for the very interesting discussions about GAP and wedderga. Vervolgens zou ik ook graag mijn collega's Andreas, Ann, Mauricio en Sara willen bedanken voor het nalezen van en hun kritische opmerkingen op een eerste versie van mijn doctoraatsthesis. Graag zou ik alle collega's van de Vrije Universiteit Brussel, en in het bijzonder Philippe, Sara, Ann, Karen en Timmy, willen bedanken voor de fijne lunches, ontspannende koffiepauzes, toffe babbels, filmavonden, wiskundige nevenactiviteiten en zoveel meer! Verder ben ik ook dank verschuldigd aan het Fonds voor Wetenschappelijk Onderzoek om mij gedurende vier jaar financieel te ondersteunen. Hierbij moet ik ook Stefaan Caenepeel bedanken om mij gedurende een jaar een assistenten- positie aan te bieden in afwachting van een aanstelling door het FWO. Mijn familie en vrienden wil ik bedanken om mij onvoorwaardelijk te steunen. Mijn laatste, maar zeker niet de minste dank gaat uit naar mijn man Giel, om zijn steun en vertrouwen. Ook een heel dikke dankjewel om dit werk van een mooie omslag te voorzien. Inneke Van Gelder maart 2015 CONTENTS contents i introduction iii summary ix publications xix samenvatting (summary in dutch) xxi list of notations xxxiii 1 preliminaries 1 1.1 Fixed point free groups . 2 1.2 Quaternion algebras . 3 1.3 Normal bases . 3 1.4 Number fields . 4 1.5 Crossed products . 10 1.6 Group rings . 11 1.7 Wedderburn-Artin decomposition . 14 1.8 Z-orders . 24 1.9 Congruence Subgroup Problem . 26 1.10 Finite subgroups of exceptional simple algebras . 31 1.11 Cyclotomic units . 33 1.12 Bass units . 34 1.13 Bicyclic units . 36 1.14 Central units . 39 2 wedderburn decomposition and idempotents 41 2.1 The Wedderburn decomposition of FG . 41 2.2 Primitive idempotents of QG ................... 55 2.3 Primitive idempotents of FG ................... 61 2.4 Conclusions . 66 i contents 3 exceptional components 67 3.1 Group algebras with exceptional components of type EC2 . 68 3.2 Group algebras with exceptional components of type EC1 . 72 3.3 Examples . 93 4 central units 101 4.1 Abelian groups . 101 4.1.1 A new proof of the Bass-Milnor Theorem . 103 4.1.2 A virtual basis of Bass units . 107 4.2 Strongly monomial groups . 110 4.3 Abelian-by-supersolvable groups . 113 4.3.1 Generalizing the Jespers-Parmenter-Sehgal Theorem . 114 4.3.2 Reducing to a basis of products of Bass units . 120 4.4 Another class within the strongly monomial groups . 124 4.5 Conclusions . 133 5 applications to units of group rings 135 5.1 A subgroup of finite index in U(Z(Cqm o1 Cpn )) . 135 5.2 A method to compute U(ZG) up to commensurability . 137 5.3 Examples . 142 + 5.3.1 U(ZD16) up to finite index . 142 5.3.2 U(ZSL(2; 5)) up to commensurability . 143 5.4 Conclusions . 144 bibliography 147 index 155 ii INTRODUCTION The notion of a group algebra already appeared in a paper of Arthur Cayley from 1854. However, only after the influential works of Richard Brauer (1901- 1977) and Emmy Noether (1882-1935) on representation theory, the subject gained attention because of the correspondence between modules of group alge- bras and group representations. In 1940, Graham Higman posed the following question in his Ph.D. thesis, for finite groups G and H: Does ZG ' ZH imply that G ' H? This problem is referred to as the (integral) isomorphism problem. It was anticipated for a long time for this conjecture to be true. In 1987, Klaus W. Roggenkamp and Leonard L. Scott showed that this indeed is the case if G is a nilpotent group. It was a surprise when Martin Hertweck gave a counter example to the isomorphism problem in his Ph.D. thesis in 1998. Nowadays, it is still an important problem to decide for which classes of groups the con- jecture does hold. In all these investigations, the unit group U(ZG) of ZG plays a fundamental role. It is essential to consider ZG as a Z-order in the (semisimple) rational group algebra QG and to have a detailed understanding of the Wedderburn decomposition of QG. If one proves the equality of two numbers a and b by showing first that `a is less than or equal to b' and then `a is greater than or equal to b', it is unfair. One should instead show that they are really equal by disclosing the inner ground for their equality | Emmy Noether The Wedderburn-Artin Theorem states that a semisimple ring R is isomor- phic to a product of finitely many ni-by-ni matrix rings over division rings Di, for some integers ni. However, in the mindset of Emmy Noether, such a classification is unfair. One should instead aim to construct an explicit isomor- phism between R and the product of matrix rings. To do this, a first important step is to calculate the primitive central idempotents e of R to distinguish the different matrix rings. Secondly, one needs to construct elements in each com- ponent Re, which play the role of a complete set of matrix units. In particular, one has to construct a complete set of orthogonal primitive idempotents. iii introduction A classical method for obtaining the primitive central idempotents of a semi- simple group algebra FG involves computations using the irreducible charac- ters of G over an algebraic closure of F . However, the known methods to com- pute the character table of a finite group are very time consuming. Therefore, in practical applications, the classical description of primitive central idempo- tents sometimes is of limited use. One would like a character-free description that only depends on the lattice of subgroups and the characteristic of the field, i.e. a description completely within FG. Such a description has been obtained by Aurora Olivieri, Angel´ del R´ıoand Juan Jacobo Sim´onin 2004 for the primitive central idempotents of QG when G is a strongly monomial group, for example an abelian-by-supersolvable group. This method relies on pairs of subgroups (H; K) of G satisfying some conditions which can be checked in- side the rational group algebra QG. Such pairs are called strong Shoda pairs of G. It turns out that each primitive central idempotent is the sum of the distinct conjugates of "(H; K) (corresponding to a natural idempotent in the rational group algebra Q(H=K)) for a strong Shoda pair (H; K), which we de- note by e(G; H; K). Furthermore, each simple component in the Wedderburn decomposition is a matrix ring over a crossed product of the finite abelian group NG(K)=H over a specific cyclotomic field for some strong Shoda pair (H; K). In 2007, Osnel Broche and Angel´ del R´ıotransfered those results to the case of semisimple finite group algebras FG for strongly monomial groups G. For arbitrary semisimple group algebras FG, it remains an open problem to give a character-free description of the primitive central idempotents and the Wedderburn decomposition of FG. For a rational group algebra QG of a finite nilpotent group G, a complete set of matrix units of an arbitrary simple component QGe(G; H; K) was given, in 2012, by Eric Jespers, Gabriela Olteanu and Angel´ del R´ıo.In joined work with Gabriela Olteanu, we gave a similar result for semisimple finite group algebras FG of nilpotent groups G. Moreover, examples were given to show that the method can not be extended to, for example, finite metacyclic groups. Chapter 1 is a preliminary chapter. In Chapter 2, we first study the primitive central idempotents and the Wedderburn decomposition of group algebras FG with F a number field and G a strongly monomial group (Theorem 2.1.6). This is a generalization of the results of Aurora Olivieri, Angel´ del R´ıo,Juan Jacobo Sim´onand Osnel Broche. Next, we focus on a complete set of matrix units in the Wedderburn components of QG and FG, with F a finite field, for a class of finite strongly monomial groups containing some metacyclic groups (Theorems 2.2.1 and 2.3.4). iv introduction I have never done anything `useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world | Godfrey Harold Hardy Regardless of his sayings, much of the work of Godfrey Harold Hardy (1877- 1947) did find applications in different branches of science, other than mathe- matics.
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