Graded Symmetry Groups: Plane and Simple

TL;DR

The paper advances a graded view of symmetry groups by showing that any product of $k$ reflections in a real geometric algebra $\mathbb{R}_{pqr}$ decomposes into exactly $\lceil k/2\rceil$ commuting simple factors, unifying Mozzi-Chasles-type decompositions across dimensions. It develops a robust invariant decomposition method to split bivectors and rotors into commuting simple components, enabling closed-form exponentials and logarithms for Spin/Pin groups and providing efficient tangent-space representations. The authors champion a plane-based geometric-algebra framework, embedding reflections and geometry in blades, and derive a Clifford representation that yields efficient matrix-vector and block-structured matrix forms of common transformations (notably $E(3)$). The results offer both conceptual clarity and computational benefits, linking geometry with Lie-group representations and extending classic theorems to broader metric signatures and higher dimensions with potential applications in physics and computer graphics.

Abstract

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 $\mathbb{R}_{pqr}$, 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.

Figures (11)

• Figure 1: Reflections in 2D. From left to right, a single line reflection, two intersecting line reflections forming a rotation, and two parallel line reflections forming a translation.
• Figure 2: Three different sets of line reflections that intersect in the same point and at the same angle create the same rotation around that point.
• Figure 3: When viewed in 3D, translations (right) can be seen as rotations (left) around infinite points.
• Figure 4: (left) Three line reflections produce a glide reflection. \ref{['ex:glide_reflection']} shows that in $\text{E}({2})$ the seemingly distinct rotoreflections are also just glide reflections. (right) Four line reflections produce no new isometry, but instead again a rotation/translation.
• Figure 5: Given four line reflections $abcd$, the gauge symmetry allows the pairs $ab$ and $cd$ to be rotated into $a'b'$ and $c' d'$, until $b'$ and $c'$ are incident and vanish from the expression. The result is a bireflection.

Theorems & Definitions (23)

• Theorem 1: invariant decomposition
• Theorem 2: Cartan-Dieudonné
• proof
• definition 1: Graded symmetry group
• Theorem 3
• proof
• Example 6.1: Invariant decomposition in STA, 3DCGA, 3DPGA.
• Example 6.2: Mozzi–Chasles' theorem in 3DPGA
• Example 6.3: Invariant decomposition in Spacetime Algebra (STA)
• Example 6.4: Seeming counter example