Symmetric bases for finite element exterior calculus spaces

In 2006, Arnold, Falk, and Winther developed finite element exterior calculus, using the language of differential forms to generalize the Lagrange, Raviart--Thomas, Brezzi--Douglas--Marini, and N\'ed\'elec finite element spaces for simplicial triangulations. In a paper, Licht asks whether, on a single simplex, one can construct bases for these spaces that are invariant with respect to permuting the vertices of the simplex. For scalar fields, standard bases all have this symmetry property, but for vector fields, this question is more complicated: such invariant bases may or may not exist, depending on the polynomial degree of the element. In dimensions two and three, Licht constructs such invariant bases for certain values of the polynomial degree $r$, and he conjectures that his list is complete, that is, that no such basis exists for other values of $r$. In this paper, we show that Licht's conjecture is true in dimension two. However, in dimension three, we show that Licht's ideas can be extended to give invariant bases for many more values of $r$; we then show that this new larger list is complete. Along the way, we develop a more general framework for the geometric decomposition ideas of Arnold, Falk, and Winther.