# Graded Symmetry Groups: Plane and Simple

arXiv: https://arxiv.org/abs/2107.03771 · HTML: https://mlsanity.com/paper/2107.03771/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jul 2021.

## Built on by (most cited first, 11 of 11)

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- [On Lie Groups Preserving Subspaces of Degenerate Clifford Algebras](https://mlsanity.com/paper/2601.07191/) (2026)
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## Abstract (arXiv)

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 graded structure on Lie groups, which is not evident in their matrix representation. By embracing this graded structure, the invariant decomposition theorem was proven: 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 function for all Spin groups, and identifies element of geometry such as planes, lines, points, as the invariants of k-reflections. We conclude by presenting novel matrix/vector representations for geometric algebras ℝpqr, and use this in E(3) to illustrate the relationship with the classic covariant, contravariant and adjoint representations for the transformation of points, planes and lines.