We study symmetries of bases and spanning sets in finite element exterior calculus, using representation theory. We want to know which vector-valued finite element spaces have bases invariant under permutation of vertex indices. The permutations of vertex indices correspond to the symmetry group of the simplex. That symmetry group is represented on simplicial finite element spaces by the pullback action. We determine a natural notion of invariance and sufficient conditions on the dimension and polynomial degree for the existence of invariant bases. We conjecture that these conditions are necessary too. We utilize Djoković and Malzan’s classification of monomial irreducible representations of the symmetric group and show new symmetries of the geometric decomposition and canonical isomorphisms of the finite element spaces. Explicit invariant bases with complex coefficients are constructed in dimensions two and three for different spaces of finite element differential forms.

我们使用表示论研究有限元外微积分中基和生成集的对称性。我们想知道哪些向量值有限元空间在顶点索引排列下具有基不变。顶点索引的排列对应于单纯形的对称群。该对称群通过回拉作用在单纯有限元空间上表示。我们确定了不变性的自然概念以及不变基存在的维数和多项式次数的充分条件。我们推测这些条件也是必要的。我们利用德约科维奇和马尔赞对对称群的单项式不可约表示的分类，并展示了有限元空间的几何分解和规范同构的新对称性。对于有限元微分形式的不同空间，在二维和三维中构造具有复系数的显式不变基。