Graded Symmetry Groups: Plane and Simple

The symmetries described by Pin groups are the result of combining a finite number of discrete reflections in (hyper)planes. The current work shows how an analysis using geometric algebra provides a picture complementary to that of the classic matrix Lie algebra approach, while retaining information about the number of reflections in a given transformation. This imposes a type of graded structure on Lie groups, not evident in their matrix representation. Embracing this graded structure, we prove the invariant decomposition theorem: any composition of k linearly independent reflections can be decomposed into \(\lceil {k/2}{\rceil }\) commuting factors, each of which is the product of at most two reflections. This generalizes a conjecture by M. Riesz, and has e.g. the Mozzi–Chasles’ theorem as its 3D Euclidean special case. To demonstrate its utility, we briefly discuss various examples such as Lorentz transformations, Wigner rotations, and screw transformations. The invariant decomposition also directly leads to closed form formulas for the exponential and logarithmic functions for all Spin groups, and identifies elements of geometry such as planes, lines, points, as the invariants of k-reflections. We conclude by presenting a novel algorithm for the construction of matrix/vector representations for geometric algebras \({\mathbb {R}}^{{}}_{pqr}\), and use this in \(\text {E}({3})\) to illustrate the relationship with the classic covariant, contravariant and adjoint representations for the transformation of points, planes and lines.