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 recent 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.

翻译：2006年，Arnold、Falk和Winther提出了有限元外微积分方法，利用微分形式的语言推广了单纯形三角剖分上的Lagrange、Raviart-Thomas、Brezzi-Douglas-Marini和Nédélec有限元空间。在近期的一篇论文中，Licht提出了一个问题：在单个单纯形上，能否构造出关于顶点置换保持不变的这些空间的基？对于标量场，所有标准基均具有该对称性；但对于向量场，问题更为复杂——此类不变基是否存在取决于多项式的次数。在二维和三维情形下，Licht针对某些多项式次数r构造了不变基，并推测其列表是完备的，即其他r值下不存在此类基。本文证明，在二维情形下Licht的猜想成立。然而在三维情形下，我们展示Licht的思想可被推广，从而为更多r值构造不变基；进而证明这一更大的新列表是完备的。在此过程中，我们发展了Arnold、Falk和Winther几何分解思想的更一般性框架。