Computing the Wedderburn decomposition of group algebras by the Brauer-Witt theorem
Author:
Gabriela Olteanu
Journal:
Math. Comp. 76 (2007), 1073-1087
MSC (2000):
Primary 20C15; Secondary 16S34
DOI:
https://doi.org/10.1090/S0025-5718-07-01957-6
Published electronically:
January 4, 2007
MathSciNet review:
2291851
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We present an alternative constructive proof of the Brauer-Witt theorem using the so-called strongly monomial characters that gives rise to an algorithm for computing the Wedderburn decomposition of semisimple group algebras of finite groups.
- [Bro-Kon-Oli-Olt-Río]
O. Broche Cristo, A. Konovalov, A. Olivieri, G. Olteanu and Á. del Río, Wedderga - Wedderburn Decomposition of Group Algebras, Version 4.0; 2006
(http://www.um.es/adelrio/wedderga.htm)
. - [Bro-Río] O. Broche Cristo and Á. del Río, Wedderburn decomposition of finite group algebras, Finite Fields Appl. 13 (2007), 71-79.
- [Cur-Rei] Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962. MR 0144979
- [GAP]
The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.4; 2006,
(http://www.gap-system.org)
. - [Her96] Allen Herman, A constructive Brauer-Witt theorem for certain solvable groups, Canad. J. Math. 48 (1996), no. 6, 1196–1209. MR 1426900, https://doi.org/10.4153/CJM-1996-063-1
- [Her97] Allen Herman, On the automorphism groups of rational group algebras of metacyclic groups, Comm. Algebra 25 (1997), no. 7, 2085–2097. MR 1451679, https://doi.org/10.1080/00927879708825973
- [Her03] Allen Herman, Using 𝐺-algebras for Schur index computation, J. Algebra 260 (2003), no. 2, 463–475. MR 1967308, https://doi.org/10.1016/S0021-8693(02)00577-X
- [Jes-Lea] Eric Jespers and Guilherme Leal, Generators of large subgroups of the unit group of integral group rings, Manuscripta Math. 78 (1993), no. 3, 303–315. MR 1206159, https://doi.org/10.1007/BF02599315
- [Jes-Lea-Paq] Eric Jespers, Guilherme Leal, and Antonio Paques, Central idempotents in the rational group algebra of a finite nilpotent group, J. Algebra Appl. 2 (2003), no. 1, 57–62. MR 1964765, https://doi.org/10.1142/S0219498803000398
- [Jes-Río] Eric Jespers and Angel del Río, A structure theorem for the unit group of the integral group ring of some finite groups, J. Reine Angew. Math. 521 (2000), 99–117. MR 1752297, https://doi.org/10.1515/crll.2000.032
- [Oli-Río] Aurora Olivieri and Ángel del Río, An algorithm to compute the primitive central idempotents and the Wedderburn decomposition of a rational group algebra, J. Symbolic Comput. 35 (2003), no. 6, 673–687. MR 1981041, https://doi.org/10.1016/S0747-7171(03)00035-X
- [Oli-Río-Sim04] Aurora Olivieri, Ángel del Río, and Juan Jacobo Simón, On monomial characters and central idempotents of rational group algebras, Comm. Algebra 32 (2004), no. 4, 1531–1550. MR 2100373, https://doi.org/10.1081/AGB-120028797
- [Oli-Río-Sim06] A.A. Olivieri, Á. del Río and J. J. Simón The group of automorphisms of a rational group algebra of a finite metacyclic group, Comm. Algebra 34 (2006), no. 10, 3543-3567.
- [Pas] Donald S. Passman, Infinite crossed products, Pure and Applied Mathematics, vol. 135, Academic Press, Inc., Boston, MA, 1989. MR 979094
- [Pie] Richard S. Pierce, Associative algebras, Graduate Texts in Mathematics, vol. 88, Springer-Verlag, New York-Berlin, 1982. Studies in the History of Modern Science, 9. MR 674652
- [Ple-Huf] V.S. Pless and W.C. Huffman, Handbook of Coding Theory, Elsevier, New York, 1998.
- [Rei] I. Reiner, Maximal orders, London Mathematical Society Monographs. New Series, vol. 28, The Clarendon Press, Oxford University Press, Oxford, 2003. Corrected reprint of the 1975 original; With a foreword by M. J. Taylor. MR 1972204
- [Río-Rui] Ángel del Río and Manuel Ruiz, Computing large direct products of free groups in integral group rings, Comm. Algebra 30 (2002), no. 4, 1751–1767. MR 1894041, https://doi.org/10.1081/AGB-120013213
- [Rit-Seh] Jürgen Ritter and Sudarshan K. Sehgal, Construction of units in integral group rings of finite nilpotent groups, Trans. Amer. Math. Soc. 324 (1991), no. 2, 603–621. MR 987166, https://doi.org/10.1090/S0002-9947-1991-0987166-9
- [Seh] S.K. Sehgal, Units of integral group rings, Longman Scientific and Technical Essex, 1993.
- [Shi] M. Shirvani, The structure of simple rings generated by finite metabelian groups, J. Algebra 169 (1994), no. 3, 686–712. MR 1302112, https://doi.org/10.1006/jabr.1994.1304
- [Spi] Karlheinz Spindler, Abstract algebra with applications. Vol. I, Marcel Dekker, Inc., New York, 1994. Vector spaces and groups. MR 1243416
- [Wei] Edwin Weiss, Cohomology of groups, Pure and Applied Mathematics, Vol. 34, Academic Press, New York-London, 1969. MR 0263900
- [Yam] Toshihiko Yamada, The Schur subgroup of the Brauer group, Lecture Notes in Mathematics, Vol. 397, Springer-Verlag, Berlin-New York, 1974. MR 0347957
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Additional Information
Gabriela Olteanu
Affiliation:
Department of Mathematics and Computer Science, North University of Baia Mare, Victoriei 76, 430072 Baia Mare, Romania.
Address at time of publication:
Department of Mathematics, University of Murcia, 30100 Murcia, Spain.
Email:
golteanu@um.es, olteanu@math.ubbcluj.ro
DOI:
https://doi.org/10.1090/S0025-5718-07-01957-6
Keywords:
Wedderburn decomposition,
Brauer--Witt theorem,
Schur index
Received by editor(s):
February 20, 2006
Published electronically:
January 4, 2007
Additional Notes:
The author was partially supported by the D.G.I. of Spain and Fundación Séneca of Murcia
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.