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Computing isometry groups of Hermitian maps


Authors: Peter A. Brooksbank and James B. Wilson
Journal: Trans. Amer. Math. Soc. 364 (2012), 1975-1996
MSC (2010): Primary 20G40
Published electronically: November 17, 2011
MathSciNet review: 2869196
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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.


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Additional Information

Peter A. Brooksbank
Affiliation: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvannia 17837
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
Email: wilson@math.ohio-state.edu, jwilson@math.colostate.edu

DOI: https://doi.org/10.1090/S0002-9947-2011-05388-2
Keywords: $*$-algebra, Hermitian map, classical group, isometry group, polynomial-time algorithm
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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.