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 Nédélec 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.

2006 年，Arnold、Falk 和 Winther 开发了有限元外微积分，使用微分形式语言推广了用于单纯三角剖分的 Lagrange、Raviart-Thomas、Brezzi-Douglas-Marini 和 Nédélec 有限元空间。在最近的一篇论文中，利希特询问，在单个单纯形上，人们是否可以为这些空间构造基，这些空间在排列单纯形的顶点方面是不变的。对于标量场，标准基都具有这种对称性，但对于向量场，这个问题更加复杂：这种不变基可能存在也可能不存在，具体取决于元素的多项式次数。在第二维和第三维中，Licht 为多项式次数r的某些值构造了这样的不变基，并且他推测他的列表是完整的，也就是说，对于r的其他值不存在这样的基。在本文中，我们证明了利希特猜想在第二维上是正确的。然而，在第三维度中，我们表明 Licht 的思想可以扩展为更多r值提供不变基；然后我们表明这个新的更大的列表是完整的。在此过程中，我们为 Arnold、Falk 和 Winther 的几何分解思想开发了一个更通用的框架。