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Computer condensation of modular representations. (English) Zbl 0705.20011

This paper contains a general account on the ideas of computer condensation of modular representations, a method to be used in order to construct large matrix representations of finite groups G over fields F with characteristic \(p>0\). (The first condensation programs were written by Parker and Thackray in 1979.)
Let V be a finitely generated FG-module. Let H be a \(p'\)-subgroup of G, and let \(e=(1/| H|)\sum_{h\in H}h\). Then \(e=e^ 2\neq 0\), and eFGe is the endomorphism ring of the permutation module \((1_ H)^ G\). This ring is also called Hecke algebra of H. Let \(\tilde V=Ve\). Then \(\tilde V\) is a finitely generated (eFGe)-module. Among other subsidiary results the author states in section 2 the following well known results from ring theory. If V is an irreducible FG-module, then \(\tilde V=Ve\) is either irreducible or zero as an (eFGe)-module (Lemma 4). Furthermore, for every (eFGe)-submodule X of \(\tilde V\) there is an FG-submodule W of V such that \(X=We\) (Lemma 3).
The condensation algorithm is based on this module correspondence. Since in general \(\tilde V=Ve\) has a smaller F-dimension, Parker’s meat-axe algorithm may be applied to find the irreducible constituents of the (eFGe)-module \(\tilde V,\) at least in favourable circumstances. The author gives a complete description of his new version of the condensation algorithm. He stresses in the introduction that condensation can only be applied to special FG-modules V for which the group action can be specified by a compact formula. He mentions that there is a program which condenses permutation modules and another program which condenses exterior powers of small matrix representations. Using these algorithms the author considers the group \(G_ 2(3)\) in characteristic 2. The actual running times of the computations are very impressive.
Reviewer: G.Michler

MSC:

20C20 Modular representations and characters
68W30 Symbolic computation and algebraic computation
20C40 Computational methods (representations of groups) (MSC2010)
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
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References:

[1] Conway, J. H.; Curis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., (An ATLAS of Finite Groups (1985), Clarendon Press: Clarendon Press Oxford) · Zbl 0568.20001
[2] Herstein, I. N., Noncommutative Rings, Carus Mathematical Monographs (1968) · Zbl 0177.05801
[3] Parker, R. A., The computer calculation of modular characters (the MEAT-AXE), (Atkinson, M. D., Computational Group Theory (1984), Academic Press: Academic Press London), 207-274 · Zbl 0555.20001
[4] Ryba, A. J.E.; Smith, S. D.; Yoshiara, S., Some projective modules determined by sporadic geometries, J. Alg. (1990), in press · Zbl 0693.20015
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