  The magnetic structure is crucial in determining the physical properties
inherent in magnetic compounds. We present an adequate descriptor for magnetic
structure with proper magnetic symmetry and high discrimination performance,
which does not depend on artificial choices for coordinate origin, axis, and
magnetic unit cell in crystal. We extend the formalism called ``smooth overlap
of atomic positions'' (SOAP), providing a numerical representation of atomic
configurations to that of magnetic moment configurations. We introduce the
descriptor in terms of the vector spherical harmonics to describe a magnetic
moment configuration and partial spectra from the expansion coefficients. We
discuss that the lowest-order partial spectrum is insufficient to discriminate
the magnetic structures with different magnetic anisotropy, and a higher-order
partial spectrum is required in general to differentiate detailed magnetic
structures on the same atomic configuration. We then introduce the fourth-order
partial spectrum and evaluate the discrimination performance for different
magnetic structures, mainly focusing on the difference in magnetic symmetry.
The modified partial spectra that are defined not to reflect the difference of
magnetic anisotropy are also useful in evaluating magnetic structures obtained
from the first-principles calculations performed without spin-orbit coupling.
We apply the present method to the symmetry-classified magnetic structures for
the crystals of Mn$_3$Ir and Mn$_3$Sn, which are known to exhibit anomalous
transport under the antiferromagnetic order, and examine the discrimination
performance of the descriptor for different magnetic structures on the same
crystal.
