The crystallographic spin groups are exact symmetries at zero SOC. Despite the spin point groups being classified six decades ago, their irreducible co-representations (co-irreps) remained elusive. Until now! The non-collinear spin point groups have co-irreps isomorphic to some magnetic point group, while the collinear spin point groups have genuinely new co-irreps. These new co-irreps are derived and tabulated. As of September 2026, our tables have been published on the Bilbao Crystallographic Server.

Collinear altermagnets are often discussed in their ideal, spin-orbit free limit. Real materials, however, always possess a finite degree of spin-orbit coupling. Here we bridge the Landau theories at zero and at finite spin orbit coupling, to predict measurable properties of real altermagnets. Along the way, we also classify all 3D crystal structures able to host collinear altermagnetism (that preserves the crystalline unit cell).

The microscopic origin of the anomalous Hall effect in CoNb3S6 has been a long-standing puzzle, partially due to the limits of neutron diffraction resolution. By using fine-momentum resolution resonant elastic x-ray scattering (REXS), we uncover a previously undetected double-Q (2Q) magnetic order featuring a long-wavelength helical modulation, that creates a staggered stripe pattern of scalar spin chirality. We provide the symmetry arguments proving how this order enables a Hall response and demonstrate how it is naturally driven by four-spin exchange interactions.