An involution is a function that is its own inverse, or equivalently \(f(f(x))=x\). There are several important involutions on Clifford objects; these commute past the grade operator with \(f(\left\langle A\right\rangle_r)=\left\langle f(A)\right\rangle_r\) and are linear: \(f(\alpha A+\beta B)=\alpha f(A)+\beta f(B)\).
The dual is documented here for convenience, even though it is not an involution (applying the dual four times is the identity).
The reverse \(A^\sim\) is given by
rev() (both Perwass and Dorst use a tilde, as in
\(\tilde{A}\) or \(A^\sim\). However, both
Hestenes and Chisholm use a dagger, as in
\(A^\dagger\). This page uses Perwass's notation).
The reverse of a term written as a product of basis vectors is
simply the product of the same basis vectors but written in reverse
order. This changes the sign of the term if the number of basis
vectors is 2 or 3 (modulo 4). Thus, for example,
\(\left(e_1e_2e_3\right)^\sim=e_3e_2e_1=-e_1e_2e_3\)
and
\(\left(e_1e_2e_3e_4\right)^\sim=e_4e_3e_2e_1=+e_1e_2e_3e_4\).
Formally, if \(X=e_{i_1}\ldots e_{i_k}\), then
\(\tilde{X}=e_{i_k}\ldots e_{i_1}\).
$$\left\langle A^\sim\right\rangle_r=\widetilde{\left\langle A\right\rangle_r}=(-1)^{r(r-1)/2}\left\langle A\right\rangle_r $$
Perwass shows that \(\left\langle AB\right\rangle_r=(-1)^{r(r-1)/2}\left\langle\tilde{B}\tilde{A}\right\rangle_r \)
The Conjugate \(A^\dagger\) is given by
Conj() (we use Perwass's notation, def 2.9 p59). This
depends on the signature of the Clifford algebra; see
grade.Rd for notation. Given a basis blade
\(e_\mathbb{A}\) with \(\mathbb{A}\subseteq\left\lbrace
1,\ldots,p+q\right\rbrace\), then we have
\(e_\mathbb{A}^\dagger = (-1)^m {e_\mathbb{A}}^\sim\),
where \(m=\mathrm{gr}_{-}(\mathbb{A})\). Alternatively,
we might say $$\left(\left\langle
A\right\rangle_r\right)^\dagger=(-1)^m(-1)^{r(r-1)/2}\left\langle
A\right\rangle_r $$ where
\(m=\mathrm{gr}_{-}(\left\langle A\right\rangle_r)\) [NB
I have changed Perwass's notation].
The main (grade) involution or grade involution
\(\widehat{A}\) is given by gradeinv(). This
changes the sign of any term with odd grade:
$$\widehat{\left\langle A\right\rangle_r} =(-1)^r\left\langle A\right\rangle_r$$
(I don't see this in Perwass or Hestenes; notation follows Hitzer and Sangwine). It is a special case of grade negation.
The grade \(r\)-negation
\(A_{\overline{r}}\) is given by neg(). This
changes the sign of the grade \(r\) component of \(A\). It is
formally defined as \(A-2\left\langle
A\right\rangle_r\) but function neg() uses a more
efficient method. It is possible to negate all terms with specified
grades, so for example we might have \(\left\langle
A\right\rangle_{\overline{\left\lbrace 1,2,5\right\rbrace}} =
A-2\left( \left\langle A\right\rangle_1 +\left\langle
A\right\rangle_2+\left\langle A\right\rangle_5\right)\) and
the R idiom would be neg(A,c(1,2,5)). Note that Hestenes
uses “\(A_{\overline{r}}\)” to mean the same as
\(\left\langle A\right\rangle_r\).
The Clifford conjugate \(\overline{A}\) is
given by cliffconj(). It is distinct from conjugation
\(A^\dagger\), and is defined in Hitzer and Sangwine as
$$\overline{\left\langle A\right\rangle_r} = (-1)^{r(r+1)/2}\left\langle A\right\rangle_r.$$
The dual \(C^*\) of a clifford object \(C\) is
given by dual(C,n); argument n is the dimension of the
underlying vector space. Perwass gives
\(C^*=CI^{-1}\)
where \(I=e_1e_2\ldots e_n\) is the unit pseudoscalar [note that Hestenes uses \(I\) to mean something different]. The dual is sensitive to the signature of the Clifford algebra and the dimension of the underlying vector space.
# S3 method for clifford
rev(x)
# S3 method for clifford
Conj(z)
cliffconj(z)
neg(C,n)
gradeinv(C)
Clifford object
Integer vector specifying grades to be negated in neg()
Robin K. S. Hankin
grade
x <- rcliff()
x
rev(x)
A <- rblade(g=3)
B <- rblade(g=4)
rev(A %^% B) == rev(B) %^% rev(A) # should be TRUE
rev(A * B) == rev(B) * rev(A) # should be TRUE
options(maxdim=8)
a <- rcliff(d=8)
dual(dual(dual(dual(a,8),8),8),8) == a # should be TRUE
options(maxdim=NULL) # restore default
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