×

An algorithm to compute the primitive central idempotents and the Wedderburn decomposition of a rational group algebra. (English) Zbl 1050.16015

The authors explain an algorithm that computes the primitive central idempotents of the rational group algebras \(\mathbb{Q} G\) for many finite groups \(G\), including Abelian-by-supersolvable groups. They have implemented this algorithm in a package of programs for System GAP, version 4, and also present an experimental comparison of the speed of this algorithm with the classical methods.

MSC:

16S34 Group rings
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
68W30 Symbolic computation and algebraic computation
16Z05 Computational aspects of associative rings (general theory)

Software:

wedderga; GAP
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baum, U.; Clausen, M., Computing irreducible representations of supersolvable groups, Math. Comp, 207, 351-359 (1994) · Zbl 0830.20031
[2] Conlon, S. B., Calculating characters of p-groups, J. Symbolic Comput, 9, 5 and 6, 535-550 (1990) · Zbl 0741.20003
[3] del Rı́o, Á.; Ruiz, M., Computing large direct products of free groups in integral group rings, Comm. Algebra, 30, 4, 1751-1767 (2002) · Zbl 1001.16018
[4] The GAP Group, 2002. GAP—Groups, Algorithms, and Programming, version 4.3. Available from http://www.gap-system.org; The GAP Group, 2002. GAP—Groups, Algorithms, and Programming, version 4.3. Available from http://www.gap-system.org
[5] Herman, A., On the automorphism group of rational group algebras of metacyclic groups, Comm. Algebra, 25, 7, 2085-2097 (1997) · Zbl 0881.20003
[6] Jespers, E.; del Rı́o, Á., A structure theorem for the unit group of the integral group ring of some finite groups, J. Reine Angew Math, 521, 99-117 (2000) · Zbl 0951.16011
[7] Jespers, E.; Leal, G., Generators of large subgroups of the unit group of integral group rings, Manuscripta Math, 78, 303-315 (1993) · Zbl 0802.16025
[8] Jespers, E., Leal, G., Paques, A., Central idempotents in rational group algebras of finite nilpotent groups. J. Algebra Appl. (in press); Jespers, E., Leal, G., Paques, A., Central idempotents in rational group algebras of finite nilpotent groups. J. Algebra Appl. (in press) · Zbl 1064.20003
[9] Olivieri, A., del Rı́o, Á., wedderga, A GAP 4 package for computing central idempotents and simple components of rational group algebras (submitted); Olivieri, A., del Rı́o, Á., wedderga, A GAP 4 package for computing central idempotents and simple components of rational group algebras (submitted)
[10] Olivieri, A., del Rı́o, Á., Simón, J.J., On monomial characters and central idempotents of rational group algebras. Comm. Algebra (in press); Olivieri, A., del Rı́o, Á., Simón, J.J., On monomial characters and central idempotents of rational group algebras. Comm. Algebra (in press)
[11] Passman, D., Infinite Crossed Products (1989), Academic Press: Academic Press Boston, MA · Zbl 0662.16001
[12] Ritter, J.; Sehgal, S. K., Construction of units in integral group rings of finite nilpotent groups, Trans. Amer. Math. Soc, 324, 603-621 (1991) · Zbl 0723.16016
[13] Sehgal, S. K., Units of Integral Group Rings (1993), Longman Scientific and Technical Essex: Longman Scientific and Technical Essex New York · Zbl 0803.16022
[14] Yamada, T., The Schur Subgroup of the Brauer Group. The Schur Subgroup of the Brauer Group, Lecture Notes in Mathematics, vol. 397 (1974), Springer-Verlag: Springer-Verlag Berlin, New York · Zbl 0321.20004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.