Computing generators of free modules over orders in group algebras II

Bley, Werner; Johnston, Henri

doi:10.1090/S0025-5718-2011-02488-9

Computing generators of free modules over orders in group algebras II
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by Werner Bley and Henri Johnston PDF

Math. Comp. 80 (2011), 2411-2434 Request permission

Abstract:

Let $E$ be a number field and $G$ a finite group. Let $\mathcal {A}$ be any $\mathcal {O}_{E}$-order of full rank in the group algebra $E[G]$ and $X$ a (left) $\mathcal {A}$-lattice. In a previous article, we gave a necessary and sufficient condition for $X$ to be free of given rank $d$ over $\mathcal {A}$. In the case that (i) the Wedderburn decomposition $E[G] \cong \bigoplus _{\chi } M_{\chi }$ is explicitly computable and (ii) each $M_{\chi }$ is in fact a matrix ring over a field, this led to an algorithm that either gives elements $\alpha _{1}, \ldots , \alpha _{d} \in X$ such that $X=\mathcal {A}\alpha _{1} \oplus \cdots \oplus \mathcal {A}\alpha _{d}$ or determines that no such elements exist. In the present article, we generalise the algorithm by weakening condition (ii) considerably.

References

Werner Bley and Robert Boltje, Computation of locally free class groups, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 4076, Springer, Berlin, 2006, pp. 72–86. MR 2282916, DOI 10.1007/11792086_{6}
Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
Werner Bley and Henri Johnston, Computing generators of free modules over orders in group algebras, J. Algebra 320 (2008), no. 2, 836–852. MR 2422318, DOI 10.1016/j.jalgebra.2008.01.042
Mark Benard and Murray M. Schacher, The Schur subgroup. II, J. Algebra 22 (1972), 378–385. MR 302747, DOI 10.1016/0021-8693(72)90155-X
Werner Bley and Stephen M. J. Wilson, Computations in relative algebraic $K$-groups, LMS J. Comput. Math. 12 (2009), 166–194. MR 2564571, DOI 10.1112/S1461157000001480
Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206, DOI 10.1007/978-3-662-02945-9
Henri Cohen, Advanced topics in computational number theory, Graduate Texts in Mathematics, vol. 193, Springer-Verlag, New York, 2000. MR 1728313, DOI 10.1007/978-1-4419-8489-0
Jean Cougnard, Un anneau d’entiers stablement libre et non libre, Experiment. Math. 3 (1994), no. 2, 129–136 (French, with English and French summaries). MR 1313877, DOI 10.1080/10586458.1994.10504284
Jean Cougnard, Anneaux d’entiers stablement libres sur $\Bbb Z[H_8\times C_2]$, J. Théor. Nombres Bordeaux 10 (1998), no. 1, 163–201 (French, with English and French summaries). MR 1827291, DOI 10.5802/jtnb.224
Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR 632548
Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. II, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. With applications to finite groups and orders; A Wiley-Interscience Publication. MR 892316
W. Eberly, Computations for algebras and group representations, Ph.D. thesis, University of Toronto, 1989.
Claus Fieker, Minimizing representations over number fields. II. Computations in the Brauer group, J. Algebra 322 (2009), no. 3, 752–765. MR 2531221, DOI 10.1016/j.jalgebra.2009.05.009
C. Friedrichs, Berechnung von Maximalordnungen über Dedekindringen, Ph.D. thesis, Technische Universität Berlin, 2000.
Emmanuel Hallouin and Christian Maire, Cancellation in totally definite quaternion algebras, J. Reine Angew. Math. 595 (2006), 189–213. MR 2244802, DOI 10.1515/CRELLE.2006.048
Ernst Kleinert, Units of classical orders: a survey, Enseign. Math. (2) 40 (1994), no. 3-4, 205–248. MR 1309127
Markus Kirschmer and John Voight, Algorithmic enumeration of ideal classes for quaternion orders, SIAM J. Comput. 39 (2010), no. 5, 1714–1747. MR 2592031, DOI 10.1137/080734467
Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2008. MR 2392026, DOI 10.1007/978-3-540-37889-1
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
Richard G. Swan, Strong approximation and locally free modules, Ring theory and algebra, III (Proc. Third Conf., Univ. Oklahoma, Norman, Okla., 1979) Lecture Notes in Pure and Appl. Math., vol. 55, Dekker, New York, 1980, pp. 153–223. MR 584612
Richard G. Swan, Projective modules over binary polyhedral groups, J. Reine Angew. Math. 342 (1983), 66–172. MR 703486, DOI 10.1515/crll.1983.342.66