Computing the Wedderburn decomposition of group algebras by the Brauer–Witt theorem
HTML articles powered by AMS MathViewer
- by Gabriela Olteanu;
- Math. Comp. 76 (2007), 1073-1087
- DOI: https://doi.org/10.1090/S0025-5718-07-01957-6
- Published electronically: January 4, 2007
- PDF | Request permission
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.References
- 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).
- O. Broche Cristo and Á. del Río, Wedderburn decomposition of finite group algebras, Finite Fields Appl. 13 (2007), 71–79.
- 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, Inc.), New York-London, 1962. MR 144979
- The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4; 2006, (http://www.gap-system.org).
- Allen Herman, A constructive Brauer-Witt theorem for certain solvable groups, Canad. J. Math. 48 (1996), no. 6, 1196–1209. MR 1426900, DOI 10.4153/CJM-1996-063-1
- Allen Herman, On the automorphism groups of rational group algebras of metacyclic groups, Comm. Algebra 25 (1997), no. 7, 2085–2097. MR 1451679, DOI 10.1080/00927879708825973
- Allen Herman, Using $G$-algebras for Schur index computation, J. Algebra 260 (2003), no. 2, 463–475. MR 1967308, DOI 10.1016/S0021-8693(02)00577-X
- 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, DOI 10.1007/BF02599315
- 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, DOI 10.1142/S0219498803000398
- 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, DOI 10.1515/crll.2000.032
- 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, DOI 10.1016/S0747-7171(03)00035-X
- 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, DOI 10.1081/AGB-120028797
- 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.
- Donald S. Passman, Infinite crossed products, Pure and Applied Mathematics, vol. 135, Academic Press, Inc., Boston, MA, 1989. MR 979094
- Richard S. Pierce, Associative algebras, Studies in the History of Modern Science, vol. 9, Springer-Verlag, New York-Berlin, 1982. Graduate Texts in Mathematics, 88. MR 674652
- V.S. Pless and W.C. Huffman, Handbook of Coding Theory, Elsevier, New York, 1998.
- 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
- 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, DOI 10.1081/AGB-120013213
- 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, DOI 10.1090/S0002-9947-1991-0987166-9
- S.K. Sehgal, Units of integral group rings, Longman Scientific and Technical Essex, 1993.
- M. Shirvani, The structure of simple rings generated by finite metabelian groups, J. Algebra 169 (1994), no. 3, 686–712. MR 1302112, DOI 10.1006/jabr.1994.1304
- Karlheinz Spindler, Abstract algebra with applications. Vol. I, Marcel Dekker, Inc., New York, 1994. Vector spaces and groups. MR 1243416
- Edwin Weiss, Cohomology of groups, Pure and Applied Mathematics, Vol. 34, Academic Press, New York-London, 1969. MR 263900
- Toshihiko Yamada, The Schur subgroup of the Brauer group, Lecture Notes in Mathematics, Vol. 397, Springer-Verlag, Berlin-New York, 1974. MR 347957
Bibliographic 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
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 76 (2007), 1073-1087
- MSC (2000): Primary 20C15; Secondary 16S34
- DOI: https://doi.org/10.1090/S0025-5718-07-01957-6
- MathSciNet review: 2291851