  Noncollinear antiferromagnets (AFMs) have recently attracted a lot of
attention owing to the potential emergence of exotic spin orders on
geometrically frustrated lattices, which can be characterized by corresponding
spin chiralities. By performing first-principles density functional
calculations together with group-theory analysis and tight-binding modelling,
here we systematically study the spin-order dependent anomalous Hall effect
(AHE) and magneto-optical effect (MOE) in representative noncollinear AFMs
Mn$_{3}X$N ($X$ = Ga, Zn, Ag, and Ni). The symmetry-related tensor shape of the
intrinsic anomalous Hall conductivity (IAHC) for different spin orders is
determined by analyzing the relevant magnetic point groups. We show that while
only the ${xy}$ component of the IAHC tensor is nonzero for right-handed spin
chirality, all other elements, $\sigma_{xy}$, $\sigma_{yz}$, and $\sigma_{zx}$,
are nonvanishing for a state with left-handed spin chirality owing to lowering
of the symmetry. Our tight-binding arguments reveal that the magnitude of IAHC
relies on the details of the band structure and that $\sigma_{xy}$ is
periodically modulated as the spin rotates in-plane. The IAHC obtained from
first principles is found to be rather large, e.g., it amounts to 359 S/cm in
Mn$_{3}$AgN. By extending our analysis to finite frequencies, we calculate the
optical isotropy
[$\sigma_{xx}(\omega)\approx\sigma_{yy}(\omega)\approx\sigma_{zz}(\omega)$] and
the magneto-optical anisotropy
[$\sigma_{xy}(\omega)\neq\sigma_{yz}(\omega)\neq\sigma_{zx}(\omega)$] of
Mn$_{3}X$N. We argue that the spin-order dependent AHE and MOE are
indispensable in detecting complex spin structures in noncollinear AFMs.
