Fundamental Automorphisms

Generally, operations like complex conjugation $\overline{AB} = \bar{A}\bar{B}$ or matrix transposition $(AB)^⊺ = B^⊺A^⊺$ are useful because they preserve or reverse multiplication. (These are called automorphisms and antiautomorphisms respectively.)

Geometric algebras possess some important automorphisms: reversion $\tilde{A}$, grade involution $A^\star$ and the combination of both, Clifford conjugation. These are useful unary operations which cannot be expressed in terms of simple geometric multiplication.

Reversion

~A or reversion(a)

Reversion $\tilde{A}$ is defined on multivectors $A$ and $B$ by the property

\[\tilde{}\,(AB) = \tilde{B}\tilde{A}\]

and by $\tilde{𝒖} = 𝒖$ for vectors. Computing the reversion looks like reversing the order of the geometric product:

julia> @basis 3[ Info: Defined basis blades v1, v2, v3, v12, v13, v23, v123, I in Main
julia> ~(v1*v2 + 2v1*v2*v3) == v2*v1 + 2v3*v2*v1true

Swapping orthogonal basis vectors $𝐯_i𝐯_j ↦ 𝐯_j𝐯_i = -𝐯_i𝐯_j$ introduces an overall factor of $-1$, and it takes $\binom{k}{2} = \frac{k(k - 1)}{2}$ swaps to reverse $k$ many basis vectors. Thus, the reversion of a homogeneous $k$-vector $A_k$ is given by

\[\tilde{A_k} = (-1)^{k(k - 1)/2} A_k\]

For inhomogeneous multivectors, reversion affects each grade separately, so the result is not always simply a change in overall sign.

Grade involution

involution(A)

Grade involution, sometimes denoted $A^\star$, is the operation of reflecting space through the origin, so that vectors are sent to their negative, $𝒖 ↦ -𝒖$. Involution is required to satisfy

\[\mathsf{involution}(AB) = \mathsf{involution}(A)\mathsf{involution}(B)\]

which means a $k$-blade of the form $B = 𝐯_1 ∧ \cdots ∧ 𝐯_k$ gets sent to $(-𝐯_1) ∧ \cdots ∧ (-𝐯_k) = (-1)^k B$. By linearity, for any $k$-vector $A$ we have

\[\mathsf{involution}(A_k) = (-1)^k A_k\]

but for inhomogeneous multivectors, involution is not always an overall change in sign.

Clifford conjugation

clifford_conj(A)

The composition of reversion and involution $\tilde{A}^\star$ is also called the Clifford conjugate.