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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\u0107 and Malzan\u2019s 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.<\/jats:p>","DOI":"10.1007\/s10208-023-09609-8","type":"journal-article","created":{"date-parts":[[2023,4,27]],"date-time":"2023-04-27T21:01:30Z","timestamp":1682629290000},"page":"1185-1224","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Symmetry and Invariant Bases in Finite Element Exterior Calculus"],"prefix":"10.1007","volume":"24","author":[{"given":"Martin W.","family":"Licht","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,4,27]]},"reference":[{"issue":"6","key":"9609_CR1","doi-asserted-by":"publisher","first-page":"3087","DOI":"10.1137\/11082539X","volume":"33","author":"M Ainsworth","year":"2011","unstructured":"Ainsworth, M., Andriamaro, G., Davydov, O.: Bernstein-B\u00e9zier finite elements of arbitrary order and optimal assembly procedures. 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