  We have analyzed galaxy and group-sized dark matter halos formed in a high
resolution LCDM numerical N-body simulation in order to study the rotation of
the triaxial figure, a property in principle independent of the angular
momentum of the particles themselves. Such figure rotation may have
observational consequences, such as triggering spiral structure in extended gas
disks. The orientation of the major axis is compared at 5 late snapshots of the
simulation. Halos with significant substructure or that appear otherwise
disturbed are excluded from the sample. We detect smooth figure rotation in 278
of the 317 halos in the sample. The pattern speeds follow a log normal
distribution centred at 0.148 h km/s/kpc with a width of 0.83. These speeds are
an order of magnitude smaller than required to explain the spiral structure of
galaxies such as NGC 2915. The axis about which the figure rotates aligns very
well with the halo minor axis, and also reasonably well with its angular
momentum vector. The pattern speed is correlated with the halo spin parameter,
but shows no correlation with the halo mass. The halos with the highest pattern
speeds show particularly strong alignment between their angular momentum
vectors and their figure rotation axes. The figure rotation is coherent outside
0.12 r_vir. The measured pattern speed and degree of internal alignment of the
figure rotation axis drops in the innermost region of the halo, which may be an
artifact of the numerical force softening. The axis ratios show a weak tendency
to become more spherical with time.
