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Graded Symmetry Groups: Plane and Simple

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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 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.

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Notes

  1. When \(|k-l|= k+l\) we have \(xy = x \wedge y.\)

  2. There are various other definition of inner products being used, such as contractions. For an overview see [9].

  3. Other conventions for \(n_o\) and \(n_{\infty }\) are in use. In fact, even demanding that \(n_o^2 = n_{\infty }^2 = 0\) and \(n_o \cdot n_{\infty } = -1,\) leaves a degree of freedom \(\alpha \) resulting in \(n_o = \tfrac{1}{2} \alpha ({\textbf{e}}_{-} - {\textbf{e}}_{+})\) and \(n_{\infty } = \alpha ^{-1} ( {\textbf{e}}_{-} + {\textbf{e}}_{+})\) [20].

  4. Note that this embedding is dual to the customary one used in CGA, where hyperspheres are instead represented by \((n-1)\)-vectors [11].

  5. For 2k-reflections with \(2k < 6\) this argument suffices; for a general proof see Sect. 5.

  6. Typically the two parameter \({{\,\textrm{arctan2}\,}}(y,x)\) function is invoked to maintain \(2\pi \) resolution, as it does all the bookkeeping needed to determine the correct quadrant. However, all such manual bookkeeping can be avoided by using Eq. (7).

  7. Note that it is important not to take away from Example 6.4 that the \(\lambda _i \in {\mathbb {C}}\) are either real or strictly imaginary: this is merely because \(B\cdot B= 0\) in Eq. (25).

  8. To convince oneself of this, consider e.g. \(B= W_1 = b_1 + b_2 + b_3,\) such that \(W_2 = b_1 b_2 + b_1 b_3 + b_2 b_3\) and \(W_{3} = b_1 b_2 b_3.\) We find that \(b_1 \wedge W_1 = b_1 b_2 + b_1 b_3\) and \(b_1 \cdot W_{3} = \lambda _1 b_2 b_3.\) Plugging this into Eq. (28), we obtain \(W_2.\)

  9. The commutator- and outer-product of a vector with a \((2m+1)\)-vector are equivalent, see e.g. [17, Chapter 1.3].

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Acknowledgements

The authors would like to thank Dr. Ir. Leo Dorst for invaluable discussions about this research. The research of M. R. was supported by KU Leuven IF project C14/16/067.

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  1. Martin Roelfs and Steven De Keninck contributed equally to this work.

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  1. Department of Physics, KU Leuven, Kortrijk, Belgium

    Martin Roelfs

  2. Informatics Institute, University of Amsterdam, Amsterdam, The Netherlands

    Steven De Keninck

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Roelfs, M., De Keninck, S. Graded Symmetry Groups: Plane and Simple. Adv. Appl. Clifford Algebras 33, 30 (2023). https://doi.org/10.1007/s00006-023-01269-9

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