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研究生： 林書愷
Lin, Shu-Kai
論文名稱： Primitive Central Idempotents in the Rational Group Algebras of Some Non-monomial Groups
Primitive Central Idempotents in the Rational Group Algebras of Some Non-monomial Groups
指導教授： 劉家新
Liu, Chia-Hsin
學位類別： 碩士
Master
系所名稱： 數學系
Department of Mathematics
論文出版年： 2020
畢業學年度： 108
語文別： 英文
論文頁數： 33
英文關鍵詞： rational group ring, primitive central idempotent, monomial group, non-monomial group
DOI URL： http://doi.org/10.6345/NTNU202000642
論文種類： 學術論文
相關次數： 點閱：103  下載：18 
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    • It is well-known that every group algebra of a finite group over the field of rational numbers is isomorphic to a direct sum of finitely many matrix rings over division rings.
    This is the so-called Wedderburn-Artin decomposition.
    It follows that there are finitely primitive central idempotents in the rational group algebra.
    However, it is not easy to write down an explicit form for each primitive central idempotent when an arbitrary group is given.
    It is known that primitive central idempotents have a nice description for finite monomial groups and nilpotent groups.
    Such description is investigated by E. Jespers, A. Olivieri and Á. del Río.
    In this thesis, we focus on some non-monomial groups and give an explicit form for primitive central idempotents.

    1 Introduction 1 2 Preliminaries 4 2.1 Notations and definitions 4 3 Construction by linear characters 7 3.1 The cyclic group from a character 7 3.2 The primitive central idempotents from linear characters 8 4 Construction by Shoda pairs 9 4.1 Shoda pairs 9 4.2 Strong Shoda pairs 11 5 The smallest non-monomial group 13 5.1 Primitive central idempotents of QA4 13 5.2 Primitive central idempotents of QSL(2, 3) 15 5.2.1 GAP code for primitive central idempotents of QSL(2, 3) 17 6 Main results 19 6.1 Primitive central idempotents of QA5 19 6.1.2 GAP code for primitive central idempotents of QA5 22 6.2 Primitive central idempotents of QSL(2, 5) 24 6.2.2 GAP code for primitive central idempotents of QSL(2, 5) 27 6.3 Primitive central idempotents of QS5 29 6.3.2 GAP code for primitive central idempotents of QS5 31 References 33

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