The central topic of the thesis is a recently proposed form of matter called altermagnetism. In common with simple antiferromagnets these are magnetic materials supporting long-range magnetic order with no net moment. In simple antiferromagnets the up and down spin electronic bands are degenerate. But in altermagnets they are not. One way of thinking about these materials is that they are nonmagnetic in real space but magnetic in momentum space thus combining features of ferromagnets and antiferromagnets. These materials have generated a great deal of interest in the spintronics community. Roughly speaking this community has, for a long time, been interested in antiferromagnets that support spin currents because antiferromagnets are insensitive to stray fields and can support faster device switching than in typical ferromagnets. Altermagnets have the potential to realize the dreams of antiferromagnetic spintronics. At the same time, altermagnets are of fundamental interest in condensed matter physics. It turns out that altermagnetism is grounded in a peculiar type of symmetry breaking described by the theory of spin groups.

The goal of this thesis project is to extend our understanding of spin groups in condensed matter especially in the direction of altermagnetism and topological materials.