Computing isometry groups of Hermitian maps

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.