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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：105  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

    [Ami55] S. A. Amitsur. Finite subgroups of division rings. Trans. Amer. Math. Soc., 80:361–386, 1955.
    [BCH+18] G. Kaur Bakshi, O. Broche Cristo, A. Herman, A. Konovalov, S. Maheshwary, A. Olivieri, G. Olteanu, Á. del Río, and I. Van Gelder. Wedderga - wedderburn decomposition of group algebras, version 4.9.5, November 2018.
    [GAP20] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.11.0, 2020.
    [Isa94] I. Martin Isaacs. Character theory of finite groups. Dover Publications, Inc., New York, 1994. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423 (57 #417)].
    [JdR16] Eric Jespers and Ángel del Río. Group ring groups. Vol. 1. Orders and generic constructions of units. De Gruyter Graduate. De Gruyter, Berlin, 2016.
    [JL01] Gordon James and Martin Liebeck. Representations and characters of groups. Cambridge University Press, New York, second edition, 2001.
    [JLP03] Eric Jespers, Guilherme Leal, and Antonio Paques. Central idempotents in the rational group algebra of a finite nilpotent group. J. Algebra Appl., 2(1):57–62, 2003.
    [OdRS04] Aurora Olivieri, Ángel del Río, and Juan Jacobo Simón. On monomial characters and central idempotents of rational group algebras. Comm. Algebra, 32(4):1531–1550, 2004.
    [vdW73a] Robert W. van der Waall. On monomial groups. J. Reine Angew. Math., 264:103–134, 1973.
    [vdW73b] Robert W. van der Waall. On the monomiality of groups of order between 100 and 200. I. J. Reine Angew. Math., 262(263):82–92, 1973.
    [vdW74] Robert W. van der Waall. On the monomiality of groups of order between 100 and 200. II. J. Reine Angew. Math., 270:184–194, 1974.

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