The discrete relations between fields and potentials with high order Whitney forms

When using the lower order Whitney elements on a simplicial complex, the matrices describing the external derivative, namely, the differential operators gradient, curl and divergence, are the incidence matrices between edges and vertices, faces and edges, tetrahedra and faces. For higher order Whitney elements, if one adopts degrees of freedom based on moments, the entries of these matrices are still equal to 0, 1 or -1 but they are no more incidence matrices. If one uses instead the weights of the field on small simplices" as alternative degrees of freedom, the matrices representative of the external derivative are incidence matrices for any polynomial degree.