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 Djokovi\'c 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.

翻译：我们利用表示论研究了有限元外微积分中基与生成集的对称性。我们旨在确定哪些向量值有限元空间具有在顶点指标置换下保持不变的基。顶点指标的置换对应于单形的对称群，该对称群通过拉回作用表示在单形有限元空间上。我们界定了不变性的自然概念，并给出了关于维度和多项式次数使得不变基存在的充分条件，同时猜想这些条件也是必要的。我们运用了Djoković与Malzan对对称群单项不可约表示的分类，并揭示了有限元空间几何分解与典范同构的新对称性。对于不同阶的有限元微分形式空间，我们在二维和三维情形构造了具有复系数的显式不变基。