Invariant decomposition

The invariant decomposition is a decomposition of the elements of pin groups ${\displaystyle {\text{Pin}}(p,q,r)}$ into orthogonal commuting elements. It is also valid in their subgroups, e.g. orthogonal, pseudo-Euclidean, conformal, and classical groups. Because the elements of Pin groups are the composition of ${\displaystyle k}$ oriented reflections, the invariant decomposition theorem reads

Every ${\displaystyle k}$-reflection can be decomposed into ${\textstyle \lceil k/2\rceil }$ commuting factors.¹

It is named the invariant decomposition because these factors are the invariants of the ${\displaystyle k}$-reflection ${\displaystyle R\in {\text{Pin}}(p,q,r)}$. A well known special case is the Chasles' theorem, which states that any rigid body motion in ${\textstyle {\text{SE}}(3)}$ can be decomposed into a rotation around, followed or preceded by a translation along, a single line. Both the rotation and the translation leave two lines invariant: the axis of rotation and the orthogonal axis of translation. Since both rotations and translations are bireflections, a more abstract statement of the theorem reads "Every quadreflection can be decomposed into commuting bireflections". In this form the statement is also valid for e.g. the spacetime algebra ${\textstyle {\text{SO}}(3,1)}$, where any Lorentz transformation can be decomposed into a commuting rotation and boost.

Any bivector ${\displaystyle F}$ in the geometric algebra ${\displaystyle \mathbb {R} _{p,q,r}}$ of total dimension ${\displaystyle n=p+q+r}$ can be decomposed into ${\displaystyle k=\lfloor n/2\rfloor }$ orthogonal commuting simple bivectors that satisfy

$${\displaystyle F=F_{1}+F_{2}\ldots +F_{k}.}$$

Defining ${\displaystyle \lambda _{i}:=F_{i}^{2}\in \mathbb {C} }$, their properties can be summarized as ${\displaystyle F_{i}F_{j}=\delta _{ij}\lambda _{i}+F_{i}\wedge F_{j}}$ (no sum). The ${\displaystyle F_{i}}$ are then found as solutions to the characteristic polynomial

$${\displaystyle 0=(F_{1}-F_{i})(F_{2}-F_{i})\cdots (F_{k}-F_{i}).}$$

Defining

$${\displaystyle W_{m}={\frac {1}{m!}}\langle F^{m}\rangle _{2m}={\frac {1}{m!}}\,\underbrace {F\wedge F\wedge \ldots \wedge F} _{m\ {\text{times}}}}$$and ${\displaystyle r=\lfloor k/2\rfloor }$, the solutions are given by

$${\displaystyle F_{i}={\begin{cases}{\dfrac {\lambda _{i}^{r}W_{0}+\lambda _{i}^{r-1}W_{2}+\ldots +W_{k}}{\lambda _{i}^{r-1}W_{1}+\lambda _{i}^{r-2}W_{3}+\ldots +W_{k-1}}}\quad &k{\text{ even}},\\[10mu]{\dfrac {\lambda _{i}^{r}W_{1}+\lambda _{i}^{r-1}W_{3}+\ldots +W_{k}}{\lambda _{i}^{r}W_{0}+\lambda _{i}^{r-1}W_{2}+\ldots +W_{k-1}}}&k{\text{ odd}}.\end{cases}}}$$

The values of ${\displaystyle \lambda _{i}}$ are subsequently found by squaring this expression and rearranging, which yields the polynomial

$${\displaystyle {\begin{aligned}0&=\sum _{m=0}^{k}\langle W_{m}^{2}\rangle _{0}(-\lambda _{i})^{k-m}\\[5mu]&=(F_{1}^{2}-\lambda _{i})(F_{2}^{2}-\lambda _{i})\cdots (F_{k}^{2}-\lambda _{i}).\end{aligned}}}$$

By allowing complex values for ${\displaystyle \lambda _{i}}$, the counter example of Marcel Riesz can in fact be solved.¹ This closed form solution for the invariant decomposition is only valid for eigenvalues ${\displaystyle \lambda _{i}}$ with algebraic multiplicity of 1. For degenerate ${\displaystyle \lambda _{i}}$ the invariant decomposition still exists, but cannot be found using the closed form solution.

A ${\displaystyle 2k}$-reflection ${\displaystyle R\in {\text{Spin}}(p,q,r)}$ can be written as ${\displaystyle R=\exp(F)}$ where ${\displaystyle F\in {\mathfrak {spin}}(p,q,r)}$ is a bivector, and thus permits a factorization

$${\displaystyle R=e^{F}=e^{F_{1}}e^{F_{2}}\cdots e^{F_{k}}.}$$

The invariant decomposition therefore gives a closed form formula for exponentials, since each ${\displaystyle F_{i}}$ squares to a scalar and thus follows Euler's formula:

$${\displaystyle R_{i}=e^{F_{i}}={\cosh }{\bigl (}{\sqrt {\lambda _{i}}}{\bigr )}+{\frac {{\sinh }{\bigl (}{\sqrt {\lambda _{i}}}{\bigr )}}{\sqrt {\lambda _{i}}}}F_{i}.}$$

Carefully evaluating the limit ${\displaystyle \lambda _{i}\to 0}$ gives

$${\displaystyle R_{i}=e^{F_{i}}=1+F_{i},}$$

and thus translations are also included.

Given a ${\displaystyle 2k}$-reflection ${\displaystyle R\in {\text{Spin}}(p,q,r)}$ we would like to find the factorization into ${\displaystyle R_{i}=\exp(F_{i})}$. Defining the simple bivector

$${\displaystyle t(F_{i}):={\frac {{\tanh }{\bigl (}{\sqrt {\lambda _{i}}}{\bigr )}}{\sqrt {\lambda _{i}}}}F_{i},}$$

where ${\displaystyle \lambda _{i}=F_{i}^{2}}$. These bivectors can be found directly using the above solution for bivectors by substituting¹

$${\displaystyle W_{m}=\langle R\rangle _{2m}{\big /}\langle R\rangle _{0}}$$

where ${\displaystyle \langle R\rangle _{2m}}$ selects the grade ${\displaystyle 2m}$ part of ${\displaystyle R}$. After the bivectors ${\displaystyle t(F_{i})}$ have been found, ${\displaystyle R_{i}}$ is found straightforwardly as

$${\displaystyle R_{i}={\frac {1+t(F_{i})}{\sqrt {1-t(F_{i})^{2}}}}.}$$

After the decomposition of ${\displaystyle R\in {\text{Spin}}(p,q,r)}$ into ${\displaystyle R_{i}=\exp(F_{i})}$ has been found, the principal logarithm of each simple rotor is given by

$${\displaystyle F_{i}={\text{Log}}(R_{i})={\begin{cases}{\dfrac {\langle R_{i}\rangle _{2}}{\textstyle {\sqrt {\langle R_{i}\rangle {\vphantom {)}}_{2}^{2}}}}}\;{\text{arccosh}}(\langle R_{i}\rangle )\quad &\lambda _{i}^{2}\neq 0,\\[5mu]\langle R_{i}\rangle _{2}&\lambda _{i}^{2}=0.\end{cases}}}$$

and thus the logarithm of ${\displaystyle R}$ is given by

$${\displaystyle {\text{Log}}(R)=\sum _{i=1}^{k}{\text{Log}}(R_{i}).}$$

So far we have only considered elements of ${\displaystyle {\text{Spin}}(p,q,r)}$, which are ${\displaystyle 2k}$-reflections. To extend the invariant decomposition to a ${\displaystyle (2k+1)}$-reflections ${\displaystyle P\in {\text{Pin}}(p,q,r)}$, we use that the vector part ${\displaystyle r=\langle P\rangle _{1}}$ is a reflection which already commutes with, and is orthogonal to, the ${\displaystyle 2k}$-reflection ${\displaystyle R=r^{-1}P=Pr^{-1}}$. The problem then reduces to finding the decomposition of ${\displaystyle R}$ using the method described above.

The bivectors ${\displaystyle F_{i}}$ are invariants of the corresponding ${\displaystyle R\in {\text{Spin}}(p,q,r)}$ since they commute with it, and thus under group conjugation

$${\displaystyle RF_{i}R^{-1}=F_{i}.}$$

Going back to the example of Chasles' theorem as given in the introduction, the screw motion in 3D leaves invariant the two lines ${\displaystyle F_{1}}$ and ${\displaystyle F_{2}}$, which correspond to the axis of rotation and the orthogonal axis of translation on the horizon. While the entire space undergoes a screw motion, these two axes remain unchanged by it.

The invariant decomposition finds its roots in a statement made by Marcel Riesz about bivectors:²

Can any bivector ${\displaystyle F}$ be decomposed into the direct sum of mutually orthogonal simple bivectors?

Mathematically, this would mean that for a given bivector ${\displaystyle F}$ in an ${\displaystyle n}$ dimensional geometric algebra, it should be possible to find a maximum of ${\textstyle k=\lfloor n/2\rfloor }$ bivectors ${\displaystyle F_{i}}$, such that ${\textstyle F=\sum _{i=1}^{\lfloor n/2\rfloor }F_{i}}$, where the ${\displaystyle F_{i}}$ satisfy ${\displaystyle F_{i}\cdot F_{j}=[F_{i},F_{j}]=0}$ and should square to a scalar ${\displaystyle \lambda _{i}:=F_{i}^{2}\in \mathbb {R} }$. Marcel Riesz gave some examples which lead to this conjecture, but also one (seeming) counter example. A first more general solution to the conjecture in geometric algebras ${\displaystyle \mathbb {R} _{n,0,0}}$ was given by David Hestenes and Garret Sobczyck.³ However, this solution was limited to purely Euclidean spaces. In 2011 the solution in ${\displaystyle \mathbb {R} _{4,1,0}}$ (3DCGA) was published by Leo Dorst and Robert Jan Valkenburg, and was the first solution in a Lorentzian signature.⁴ Also in 2011, Charles Gunn was the first to give a solution in the degenerate metric ${\displaystyle \mathbb {R} _{3,0,1}}$.⁵ This offered a first glimpse that the principle might be metric independent. Then, in 2021, the full metric and dimension independent closed form solution was given by Martin Roelfs in his PhD thesis.⁶ And because bivectors in a geometric algebra ${\displaystyle \mathbb {R} _{p,q,r}}$ form the Lie algebra ${\displaystyle {\mathfrak {spin}}(p,q,r)}$, the thesis was also the first to use this to decompose elements of ${\displaystyle {\text{Spin}}(p,q,r)}$ groups into orthogonal commuting factors which each follow Euler's formula, and to present closed form exponential and logarithmic functions for these groups. Subsequently, in a paper by Martin Roelfs and Steven De Keninck the invariant decomposition was extended to include elements of ${\displaystyle {\text{Pin}}(p,q,r)}$, not just ${\displaystyle {\text{Spin}}(p,q,r)}$, and the direct decomposition of elements of ${\displaystyle {\text{Spin}}(p,q,r)}$ without having to pass through ${\displaystyle {\mathfrak {spin}}(p,q,r)}$ was found.¹