SO(10)
In particle physics, SO(10) refers to a grand unified theory (GUT) based on the spin group Spin(10). The name SO(10) is conventional among physicists, and derives from the Lie group SO(10), which is a special orthogonal group that is double covered by Spin(10).
Before the SU(5) theory behind the Georgi–Glashow model, Harald Fritzsch and Peter Minkowski and independently Howard Georgi found that all the matter contents are incorporated into a single representation, spinorial 16 of SO(10). (Historical note: the before in the previous sentence is misleading: Georgi found the SO(10) theory a few hours before finding SU(5) at the end of 1973.)
It has the branching rules to [SU(5)×U(1)χ]/Z5.
If the hypercharge is contained within SU(5), this is the conventional Georgi–Glashow model, with the 16 as the matter fields, the 10 as the electroweak Higgs field and the 24 within the 45 as the GUT Higgs field. The superpotential may then include renormalizable terms of the form Tr(45 ⋅ 45); Tr(45 ⋅ 45 ⋅ 45); 10 ⋅ 45 ⋅ 10, 10 ⋅ 16* ⋅ 16 and 16* ⋅ 16. The first three are responsible to the gauge symmetry breaking at low energies and give the Higgs mass, and the latter two give the matter particles masses and their Yukawa couplings to the Higgs.
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