\cdbalgorithm{projweyl}{}

Projects an expression onto Weyl spinors of positive chirality (this
algorithm only works in even dimensions). On such a subspace, we have
\begin{equation}
\label{e:g10toeps}
\Gamma^{r_1 \cdots r_{d}}\Big|_{\text{Weyl}} = \frac{1}{\sqrt{-g}}\epsilon^{r_1\cdots
r_{d}}
\, ,\quad \epsilon^{0\cdots (d-1)} = +1\, ,
\end{equation}
and therefore all gamma matrices with more than $d/2$ indices can be
converted to their ``dual'' gamma matrices. By repeated contraction
of~\eqref{e:g10toeps} with gamma matrices on the left one deduces that
\begin{equation}
\Gamma^{r_1\cdots r_n}\Big|_{\text{Weyl}} = \frac{1}{\sqrt{-g}} \frac{(-1)^{\frac{1}{2}n(n+1)+1}}{(d-n)!}
\Gamma_{s_1\cdots s_{d-n}}\Big|_{\text{Weyl}} \epsilon^{s_1\cdots s_{d-n} r_1\cdots r_n}\, .
\end{equation}
Here is an example:
\begin{screen}{1,2}
{m,n,p,q,r,s,t}::Indices.
{m,n,p,q,r,s,t}::Integer(0..5).
\Gamma{#}::GammaMatrix.
\Gamma_{m n p q};
@projweyl!(%);
\end{screen}

\cdbseeprop{GammaMatrix}
\cdbseealgo{join}
