Dear community, I'm trying to manipulate matrix valued differential forms. By academic deformation, I'm used to work in models with extra dimensions (say five to analyse the simplest case).
The problem I'm having is with the order of indices and elements of the expressions.
Below the calculations:
Example of notebook
Some initial definitions
{A,B,C,L,M,N,P,Q}::Indices(full,position=independent);
{\alpha,\beta,\mu,\nu,\lambda,\rho,\sigma}::Indices(position=independent, parent=full);
d{#}::ExteriorDerivative.
d{#}::LaTeXForm("\mathrm{d}").
\delta{#}::KroneckerDelta.
G_{M N}::Metric(signature=-1).
G^{M N}::InverseMetric.
\epsilon{#}::EpsilonTensor(metric=G);
Notation and properties
eh{#}::LaTeXForm("\hat{e}").
oh{#}::LaTeXForm("\hat{\omega}").
Rh{#}::LaTeXForm("\hat{R}").
Th{#}::LaTeXForm("\hat{T}").
{eh^{A?}, oh^{A? B?}}::DifferentialForm(degree=1);
{Th^{A?}, Rh^{A? B?}}::DifferentialForm(degree=2);
{oh^{A? B?}, Rh^{A? B?}}::AntiSymmetric.
Rh^{A? B?}::SelfCommuting.
eh^{A?}::SelfAntiCommuting.
{\epsilon{#}, Rh^{4 B}, Rh^{A B}, eh^{m}, eh^{4}}::SortOrder;
Working a term
Lambda := \epsilon_{A B C L M} eh^{A} ^ eh^{B} ^ eh^{C} ^ eh^{L} ^ eh^{M};
split_index(Lambda, $M, m, 4$, repeat = True)
sort_product(_)
canonicalise(_);
The result is shown in the
Discussion
As you can see, the wedge product is not sorted properly, nor the indices!
the right result should be
$$\epsilon_{a b c l m} \hat{e}^{m}\wedge \hat{e}^{a}\wedge \hat{e}^{b}\wedge \hat{e}^{c}\wedge \hat{e}^{l} - 5\epsilon_{4 a b c l} \hat{e}^{l}\wedge \hat{e}^{a}\wedge \hat{e}^{b}\wedge \hat{e}^{c}\wedge \hat{e}^{4}.$$
Any suggestion on how to manipulate correctly these expressions?
Update: Looking at the source code I found that the algorithm indexsortis implemented, but the problem persists. It seems to be a "bug" with the sort_product and canonicalise algorithms on differential forms.