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Computing with characters of finite groups. (English) Zbl 0719.20007

This is a description of major additions in the Character Algorithm System CAS (Aachen) since the 1984 report [J. Neubüser, H. Pahlings, W. Plesken, Computational Group Theory, Proc. Symp., Durham 1982, 195-247 (1984; Zbl 0546.20001)] together with recent examples of its use in computing character-tables. These revisions include new integral bases for cyclotomic fields (to support exact arithmetic of character values), implementation of the LLL-algorithm for finding short vectors in lattices (generalized characters of small norm), and B. Fischer’s technique of Clifford matrices (to compute the character table of G from that of a factor group G/N). This report focusses on the second and third topic.
The LLL-algorithm has turned out to be the working horse of CAS. Starting with the knowledge of the conjugacy classes and the power map of a finite group G one computes the characters induced from maximal cyclic subgroups, adds the trivial character, and calls LLL to obtain a lot of irreducible characters of G together with some remaining reducible characters of small norm which then must (and often can) be handled. Examples \(J_ 1\), \(O^-_{12}(2).2\), \(M_{22}\) show the effectiveness of this approach.
The character-tables of maximal subgroups of sporadic simple groups were needed in work on the realization of simple groups as Galois groups over \({\mathbb{Q}}\) and in work on their decomposition numbers. These maximal subgroups use to have factor groups whose character tables have been computed previously. In this situation Fischer’s method of Clifford matrices has proved indispensable. The author describes the method and explains it in detail for a group \(G=2^ 4:A_ 5.\)
\(\{\) Reference [Pah 90] has appeared: Bayreuther Math. Schr. 33, 137-152 (1990; Zbl 0705.11067).\(\}\) This article is reprinted in: G. M. Piacentini Cattaneo, E. Strickland (eds.), Topics in Computational Algebra (Computational Algebra Seminar, Rome ‘Tor Vergate’, 9-11 May 1990), Kluwer Acad. Publ. (1990; Zbl 0723.00021).

MSC:

20C40 Computational methods (representations of groups) (MSC2010)
20C15 Ordinary representations and characters
20D08 Simple groups: sporadic groups
20C34 Representations of sporadic groups
20-04 Software, source code, etc. for problems pertaining to group theory
12F12 Inverse Galois theory

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References:

[1] O. Bonten, Clifford-Matrizen, Diplomarbeit Aachen 1988
[2] J. H. Conway, Character Calisthenics; in: Computational Group Theory (ed. M. D. Atkinson) pp. 249-266, London: Academic Press 1984 · Zbl 0558.20007
[3] B. Fischer, Clifford Matrizen, manuscript 1982
[4] B. Fischer, unpublished manuscript 1985
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[11] H. Pahlings, Realizing finite groups as Galois groups. To appear · Zbl 0705.11067
[12] H. Pahlings, W. Plesken, Group actions on Cartesian powers with applications to representation theory. J. reine angew. Math., 380 (1987), 178-195 · Zbl 0613.20008 · doi:10.1515/crll.1987.380.178
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