so10

SO(10) is the special orthogonal group in ten dimensions, defined as the group of 10-by-10 real matrices A that satisfy A^T A = I and det(A) = 1. It forms a compact, connected Lie group under matrix multiplication and is the Lie group associated with the real Lie algebra so(10), the set of 10-by-10 real skew-symmetric matrices.

The group has dimension 45, and its Lie algebra so(10) is a semisimple algebra of type D5 in the Cartan classification. SO(10) is simple as a Lie group, meaning it has no nontrivial connected normal Lie subgroups. The universal covering group of SO(10) is Spin(10), which is simply connected and serves as a double cover of SO(10). The fundamental representations of SO(10) include the vector representation of dimension 10 and the adjoint representation of dimension 45; the corresponding spinor representations arise from Spin(10) and come in two inequivalent complex chiral representations, each of dimension 16.

SO(10) contains several notable maximal subgroups, including SU(5) × U(1) and the Pati–Salam group SU(4) × SU(2)L × SU(2)R. It also embeds SU(5) as a subgroup, providing routes for symmetry breaking down to the Standard Model gauge group.

In physics, SO(10) grand unified theories attempt to unify the Standard Model forces within a single gauge group. A single 16-dimensional spinor representation of Spin(10) can accommodate one generation of Standard Model fermions, including a right-handed neutrino, highlighting the appeal of SO(10) in model building.