Computing isometry groups of Hermitian maps
HTML articles powered by AMS MathViewer
- by Peter A. Brooksbank and James B. Wilson PDF
- Trans. Amer. Math. Soc. 364 (2012), 1975-1996 Request permission
Abstract:
A theorem is proved on the structure of the group of isometries of a Hermitian map $b\colon V\times V\to W$, where $V$ and $W$ are vector spaces over a finite field of odd order. Also a Las Vegas polynomial-time algorithm is presented which, given a Hermitian map, finds generators for, and determines the structure of its isometry group. The algorithm can be adapted to construct the intersection over a set of classical subgroups of $\operatorname {GL}(V)$, giving rise to the first polynomial-time solution of this old problem. The approach yields new algorithmic tools for algebras with involution, which in turn have applications to other computational problems of interest. Implementations of the various algorithms in the Magma system demonstrate their practicability.References
- A. Adrian Albert, Structure of algebras, American Mathematical Society Colloquium Publications, Vol. XXIV, American Mathematical Society, Providence, R.I., 1961. Revised printing. MR 0123587
- Eva Bayer-Fluckiger, Principe de Hasse faible pour les systèmes de formes quadratiques, J. Reine Angew. Math. 378 (1987), 53–59 (French). MR 895284, DOI 10.1515/crll.1987.378.53
- 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
- Peter A. Brooksbank and E. A. O’Brien, On intersections of classical groups, J. Group Theory 11 (2008), no. 4, 465–478. MR 2429348, DOI 10.1515/JGT.2008.027
- Peter A. Brooksbank and E. A. O’Brien, Constructing the group preserving a system of forms, Internat. J. Algebra Comput. 18 (2008), no. 2, 227–241. MR 2403820, DOI 10.1142/S021819670800441X
- W. Eberly and M. Giesbrecht, Efficient decomposition of associative algebras over finite fields, J. Symbolic Comput. 29 (2000), no. 3, 441–458. MR 1751390, DOI 10.1006/jsco.1999.0308
- 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
- Bettina Eick, C. R. Leedham-Green, and E. A. O’Brien, Constructing automorphism groups of $p$-groups, Comm. Algebra 30 (2002), no. 5, 2271–2295. MR 1904637, DOI 10.1081/AGB-120003468
- Daniel Goldstein and Robert M. Guralnick, Alternating forms and self-adjoint operators, J. Algebra 308 (2007), no. 1, 330–349. MR 2290925, DOI 10.1016/j.jalgebra.2006.06.009
- Derek F. Holt and Sarah Rees, Testing modules for irreducibility, J. Austral. Math. Soc. Ser. A 57 (1994), no. 1, 1–16. MR 1279282
- Gábor Ivanyos, Fast randomized algorithms for the structure of matrix algebras over finite fields (extended abstract), Proceedings of the 2000 International Symposium on Symbolic and Algebraic Computation (St. Andrews), ACM, New York, 2000, pp. 175–183. MR 1805121, DOI 10.1145/345542.345620
- Gábor Ivanyos and Klaus Lux, Treating the exceptional cases of the MeatAxe, Experiment. Math. 9 (2000), no. 3, 373–381. MR 1795309
- Nathan Jacobson, Lectures in abstract algebra, Graduate Texts in Mathematics, No. 31, Springer-Verlag, New York-Berlin, 1975. Volume II: Linear algebra; Reprint of the 1953 edition [Van Nostrand, Toronto, Ont.]. MR 0369381
- David W. Lewis, Involutions and anti-automorphisms of algebras, Bull. London Math. Soc. 38 (2006), no. 4, 529–545. MR 2249484, DOI 10.1112/S002460930601873X
- Lajos Rónyai, Computing the structure of finite algebras, J. Symbolic Comput. 9 (1990), no. 3, 355–373. MR 1056632, DOI 10.1016/S0747-7171(08)80017-X
- Robert Steinberg, Generators for simple groups, Canadian J. Math. 14 (1962), 277–283. MR 143801, DOI 10.4153/CJM-1962-018-0
- E. J. Taft, Invariant Wedderburn factors, Illinois J. Math. 1 (1957), 565–573. MR 98124
- D. E. Taylor, Pairs of generators for matrix groups I, The Cayley Bulletin 3 (1987), 76–85.
- L. J. Rylands and D. E. Taylor, Matrix generators for the orthogonal groups, J. Symbolic Comput. 25 (1998), no. 3, 351–360. MR 1615330, DOI 10.1006/jsco.1997.0180
- A. Wagner, On the classification of the classical groups, Math. Z. 97 (1967), 66–76. MR 209366, DOI 10.1007/BF01111124
- André Weil, Algebras with involutions and the classical groups, J. Indian Math. Soc. (N.S.) 24 (1960), 589–623 (1961). MR 136682
- James B. Wilson, Decomposing $p$-groups via Jordan algebras, J. Algebra 322 (2009), no. 8, 2642–2679. MR 2559855, DOI 10.1016/j.jalgebra.2009.07.029
- James B. Wilson, Finding central decompositions of $p$-groups, J. Group Theory 12 (2009), no. 6, 813–830. MR 2582050, DOI 10.1515/JGT.2009.015
- J. B. Wilson, Optimal Gram-Schmidt algorithms (submitted). arXiv:0910.0435.
Additional Information
- Peter A. Brooksbank
- Affiliation: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvannia 17837
- MR Author ID: 321878
- Email: pbrooksb@bucknell.edu
- James B. Wilson
- Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
- Address at time of publication: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523
- MR Author ID: 881789
- Email: wilson@math.ohio-state.edu, jwilson@math.colostate.edu
- Received by editor(s): June 19, 2009
- Received by editor(s) in revised form: November 18, 2009, March 25, 2010, May 20, 2010, and May 26, 2010
- Published electronically: November 17, 2011
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 1975-1996
- MSC (2010): Primary 20G40
- DOI: https://doi.org/10.1090/S0002-9947-2011-05388-2
- MathSciNet review: 2869196