Computing the Wedderburn decomposition of group algebras by the Brauer–Witt theorem

Olteanu, Gabriela

doi:10.1090/S0025-5718-07-01957-6

Computing the Wedderburn decomposition of group algebras by the Brauer–Witt theorem
HTML articles powered by AMS MathViewer

by Gabriela Olteanu PDF

Math. Comp. 76 (2007), 1073-1087 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 0144979
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. 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 0263900
Toshihiko Yamada, The Schur subgroup of the Brauer group, Lecture Notes in Mathematics, Vol. 397, Springer-Verlag, Berlin-New York, 1974. MR 0347957