  Operator fractional Brownian motions (OFBMs) are zero mean, operator
self-similar (o.s.s.), Gaussian processes with stationary increments. They
generalize univariate fractional Brownian motions to the multivariate context.
It is well-known that the so-called symmetry group of an o.s.s. process is
conjugate to subgroups of the orthogonal group. Moreover, by a celebrated
result of Hudson and Mason, the set of all exponents of an operator
self-similar process can be related to the tangent space of its symmetry group.
  In this paper, we revisit and study both the symmetry groups and exponent
sets for the class of OFBMs based on their spectral domain integral
representations. A general description of the symmetry groups of OFBMs in terms
of subsets of centralizers of the spectral domain parameters is provided. OFBMs
with symmetry groups of maximal and minimal types are studied in any dimension.
In particular, it is shown that OFBMs have minimal symmetry groups (as thus,
unique exponents) in general, in the topological sense. Finer classification
results of OFBMs, based on the explicit construction of their symmetry groups,
are given in the lower dimensions 2 and 3. It is also shown that the
parametrization of spectral domain integral representations are, in a suitable
sense, not affected by the multiplicity of exponents, whereas the same is not
true for time domain integral representations.
