Symmetry and Invariant Bases in Finite Element Exterior Calculus

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.


Introduction
The Lagrange finite element space over a simplex is easily defined for arbitrary polynomial degree. The literature knows several canonical bases for higher-degree Lagrange spaces, such as the standard nodal bases, the barycentric bases, and the Bernstein bases [3,1,31]. A convenient feature of these canonical bases is their symmetry: the bases do not change if we re-number the vertices of the simplex. Equivalently, they are invariant under pullback along the affine automorphisms of the simplex.
While this convenient feature might easily be taken for granted, it fails to hold for vector-valued finite element spaces, such as the Raviart-Thomas spaces, Brezzi-Douglas-Marini spaces, and the Nédélec spaces of first and second kind [41,14,37]. Indeed, even finding explicit bases for these vector-valued finite element spaces is a non-trivial topic and has only been addressed after the turn of the century [8,9,26,22,11,34]. Whether an invariant basis of a given polynomial degree exists seems to be an intricate question: for example, while no such basis exists for the space of constant vector fields over a triangle, one easily finds such a basis for the linear vector fields over a triangle. What pattern emerges for higher polynomial degrees?
The purpose of this article is to address this question: we present a natural notion of invariance and study the existence of invariant bases for vector-valued finite element spaces. As we demonstrate in this article, the existence of such bases seems to depend on the polynomial degree and the dimension. We identify sufficient conditions on these parameters for different families of finite element spaces, and we conjecture that these conditions are also necessary. To the author's best knowledge, no prior contributions to this fundamental topic exist. We use a novel connection between finite element methods and group representation theory, and a recursive construction of geometrically decomposed bases that is of independent interest. In what follows, we summarize the results and their prospective applications.
We adopt the framework of finite element exterior calculus (FEEC, [8]), which translates vector-valued finite element spaces into the calculus of differential forms. FEEC provides a unifying perspective on numerous aspects of finite element theory previously only known for special cases, such as convergence estimates [10,7,6,4], approximation theory [17,28,33,32], and a posteriori error estimation [20].
The fundamental connection to representation theory is as follows. Every permutation of the vertices of a simplex corresponds to a unique affine automorphism of that simplex, and the pullback along that automorphism acts on differential forms. Associating permutations to pullbacks in this manner preserves the group-structure. Hence representation theory enters the picture naturally: the permutation group is represented by the pullback operation over differential forms.
It turns out that a satisfying theory of invariant bases involves complex coefficients. Thus our notion of invariance in this article means invariant under the action of the symmetric group up to multiplication by complex units. In the language of representation theory, we are interested under which conditions the action of the symmetric group can be represented by monomial matrices with real or complex coefficients [38]. The transition to complex numbers reveals interesting structures: for example, the constant complex vector fields over a triangle have a basis invariant up to multiplication by complex roots of unity. The language of differential forms is essential for our exposition.
At the heart of our analysis is a new recursive construction of finite element bases, which is interesting in its own right. We construct invariant bases for finite element spaces of higher polynomial degree via reduction to the case of lower polynomial degree. For that purpose, we analyze how simplicial symmetries interact with two fundamental concepts in finite element exterior calculus. On the one hand, we recall the geometric decomposition of the finite element spaces [9], We prove that the traces and extension operators commute with the pullbacks along simplicial symmetries.
In particular, the geometrically decomposed bases are invariant if the individual summands are. Therefore, we can construct geometrically decomposed invariant bases from invariant bases with boundary conditions. On the other hand, we recall the canonical isomorphisms over an n-dimensional simplex T [8], namely, These isomorphisms are natural in the sense that they preserve the canonical spanning sets [34]. We prove that they commute with the simplicial symmetries up to signs and thus preserve invariant bases. In particular, we get invariant bases for finite element spaces with boundary conditions from invariant bases for finite element spaces of generally lower polynomial degree. Combining these two ideas, we recursively construct invariant bases, using invariant bases that we have discovered in the necessary base cases.
The aforementioned base cases refer to the finite element spaces of lowest polynomial degree, that is, constant fields. Djoković and Malzan's classification of monomial irreducible representations of the symmetric group [21] shows that invariant bases for constant fields exist only in special cases: the scalar and volume forms, the constant differential forms up to dimension 3, and constant 2-forms over 4-simplices. We outline the invariant bases for constant fields, using vector calculus notation. Let us write λ i for the barycentric coordinate associated to the i-th vertex of a tetrahedron or a triangle.
Starting from these base cases, we recursively construct bases for finite element spaces of differential forms. Inspection of the base cases then reveals whether the construction yields bases invariant up to complex units or even up to signs: this generally depends on the simplex dimension and the polynomial degree.
As a convenience for the reader, we summarize the application of our theory to common (real-valued) finite element spaces below. Here, we use the language of vector analysis and the following notation: λ i is again the barycentric coordinate associated with the i-th vertex, φ ij and φ ijk are Whitney forms with the respective index sets, and A(r, n) are the multiindices of weight r in the indices {0, 1, . . . , n}; see also Section 2. The following finite element spaces have bases that are invariant up to sign changes under reordering of the vertices: • The Brezzi-Douglas-Marini space of degree r over a triangle T , • The divergence-conforming Raviart-Thomas space of degree r over a tetrahedron T , if r ∈ {0, 1, 2, 3, 4, 6, 7, 10}.
• The curl-conforming Nédélec space of the first kind of degree r over a tetrahedron T , • The curl-conforming Nédélec space of the second kind of degree r over a tetrahedron T , However, the complex-valued versions of these finite element spaces have bases invariant up to multiplication by cubic roots of unity, irrespective of the polynomial degree.
We conjecture that the above list of invariant bases is exhaustive and discuss some partial results in that regard. As we show, if an invariant basis is already geometrically decomposed, in an intuitive sense formalized in the article, then we can reverse the recursive argument. For example, a basis of P 3 Λ 1 (T ) over a triangle cannot be geometrically decomposed and invariant up to signs because otherwise we could construct a basis of P 0 Λ 1 (T ) invariant up to signs. Similarly, a geometrically decomposed basis of P r Λ 1 (T ) over a tetrahedron that is invariant up to signs gives rise to a basis of P r Λ 1 (F ) over a face F invariant up to signs. More generally, the recursive argument gives necessary conditions on bases to be both geometrically decomposed and invariant up to signs. This does not rule out the existence of bases invariant up to signs that are not geometrically decomposed.
Bases for finite element spaces have been the subject of research for a long time. Bases for vector-valued finite element spaces, such as Brezzi-Douglas-Marini spaces, Raviart-Thomas spaces, or Nédélec spaces have been stated explicitly only very recently [8,9,26,22,11,34]. The choice of bases influences the condition numbers and sparsity properties of finite element matrices [3,42,12]. The present contribution continues previous research on bases and spanning sets in finite element exterior calculus [34].
The invariance of bases under renumbering of the vertices is a fundamental aspect of finite element spaces. It is not an issue for scalar-valued finite element spaces but becomes highly nontrivial for vector-valued finite element spaces, and no prior publication systematically discusses this topic. We remark that the seminal article of Arnold, Falk, and Winther [8] utilized techniques of representation theory to classify the affinely invariant finite-dimensional vector spaces of polynomial differential forms.
Invariant bases are not only theoretically interesting but also of natural practical interest. Suppose that the triangulation has a significant share of regular triangles or tetrahedra: if the bases exhibit the same geometric symmetries, then redundancies in the matrix coefficients can further reduce the assembly time of the local high-degree finite element matrices and, perhaps more importantly, their memory footprint. Moreover, preliminary calculations indicate that invariant bases have natural orthogonality relations in such highly regular settings; see also Example 8.8. This hints at good conditioning of the bases in broader, more common settings when the triangulations are less regular. While we rely on a recursive basis construction, we notice that certain recursive structures already enable fast algorithms for finite element operators [31,30] (see also [1]), to reduce the computational complexity of higher-degree methods.
Finite element bases with complex coefficients presumably suit best where complex coefficients emerge naturally, such as numerical electromagnetism or complex-shifted Laplacian Helmholtz solvers [18]. Apart from computational studies, future research will study symmetric bases on cubical finite element spaces [4,5,25] and the interaction of simplicial symmetries with resolutions of finite element spaces [15]. Moreover, constructing symmetric degrees of freedom is a natural follow-up endeavor [2]. Some aspects of our analysis are of broader interest in representation theory. Our recursive construction showcases new aspects of the representation theory of the symmetric group. The notion of monomial representation is central to our contribution. However, monomial representations do not seem to be a standard topic in introductory textbooks on representation theory, and only few articles approach constructive aspects of monomial representations (see [39,40]). We also remark that groups of monomial matrices over finite fields have found use in cryptography and coding theory [24]. The representation theory of groups has had various applications throughout numerical and computational mathematics, such as in geometric integration theory [16,45,36] and artificial neural networks [13]. Our application of representation theory in finite element methods adds a new entry to that list.
The remainder of this work is structured as follows. Important preliminaries on combinatorics, exterior calculus, and polynomial differential forms are summarized in Section 2. We review elements of representation theory in Section 3. In Section 4 we establish first results on the coordinate transformation of polynomial differential forms. In Section 5 we study invariant bases and spanning sets for lowest-degree finite element spaces. We discuss the symmetry properties of the canonical isomorphisms in Section 6. We discuss extension operators, geometric decompositions, and their symmetry properties in Section 7. Putting these results together, Section 8 discusses the recursive construction of invariant bases.

Notation and Definitions
We introduce and review notions from combinatorics, simplicial geometry, and differential forms over simplices. Parts of this section summarize results in [34]. We refer to Arnold, Falk, and Winther [8,9] and to Hiptmair [27] for further background on polynomial differential forms.
We write A(n) for the set of multiindices over [0 : n], that is, the set of functions α : [0 : n] → N 0 . For any α ∈ A(n), and we let ⌊α⌋ denote the minimal element of [α] provided that [α] is not empty, and ⌊α⌋ := ∞ otherwise. We let A(r, n) be the set of those α ∈ A(n) for which |α| = r. The sum α + β of α, β ∈ A(n) is defined in the obvious manner. We let δ p : Z → N be the function that equals 1 at p and is zero otherwise. When α ∈ A(r, n) and p ∈ [0 : n], then α + p ∈ A(r + 1, n) is notation for α + δ p . Similarly, when p ∈ [α], then α − p ∈ A(r − 1, n) is notation for α − δ p .

2.2.
Simplices. Let n ∈ N 0 . An n-dimensional simplex T is the convex closure of pairwise distinct affinely independent points v T 0 , . . . , v T n in Euclidean space, called the vertices of T . We call F ⊆ T a subsimplex of T if the set of vertices of F is a subset of the set of vertices of T . We write ı(F, T ) : F → T for the set inclusion of F into T .
As an additional structure, we assume that the vertices of all simplices are ordered. For simplicity and without loss of generality, we assume that all simplices have vertices ordered compatibly to the order of vertices on their subsimplices. Suppose that F is an m-dimensional subsimplex of T with ordered vertices v F 0 , . . . , v F m . With a mild abuse of notation, we let ı(F, T ) ∈ Σ 0 (m, n) be the strictly ascending mapping defined by v T ı(F,T )(i) = v F i . Here, each vertex index of F is mapped to the corresponding vertex index of T .

Barycentric Coordinates and Differential
Forms. Let T be a simplex of dimension n. Following the notation of [8] and letting k ∈ Z, we denote by Λ k (T ) the space of differential k-forms over T with real coefficients whose derivatives of all orders are smooth and bounded. Recall that these mappings take values in the k-th exterior power of the dual of the tangential space of the simplex T . In the case k = 0, the space Λ 0 (T ) = C ∞ (T ) is just the space of smooth functions over T with uniformly bounded derivatives. We use that Λ k (T ) = {0} unless 0 ≤ k ≤ n without further mention. We write RΛ k (T ) = Λ k (T ) and let CΛ k (T ) denote the complexification of RΛ k (T ). All the algebraic operations defined in the following apply to CΛ k (T ) completely analogously.
The barycentric coordinates λ T 0 , . . . , λ T n ∈ Λ 0 (T ) are the unique affine functions over T that satisfy the Lagrange property The barycentric coordinate functions of T are linearly independent and constitute a partition of unity: We write dλ T 0 , dλ T 1 , . . . , dλ T n ∈ Λ 1 (T ) for the exterior derivatives of the barycentric coordinates. These are differential 1-forms and constitute a partition of zero: It can be shown that this is, up to scaling, the only linear dependence between the exterior derivatives of the barycentric coordinate functions. Several classes of differential forms over T that are expressed in terms of the barycentric coordinates and their exterior derivatives. When r ∈ N 0 and α ∈ A(r, n), then the corresponding barycentric polynomial over T is When a, b ∈ N 0 and σ ∈ Σ(a : b, 0 : n), the corresponding barycentric alternator is Here, we treat the special case σ = ∅ by defining dλ T ∅ = 1. Whenever a, b ∈ N 0 and ρ ∈ Σ(a : b, 0 : n), then the corresponding Whitney form is In the special case that ρ : [0 : n] → [0 : n] is the single member of Σ 0 (n, n), we write φ T := φ ρ for the associated Whitney form. In what follows, the polynomials (7) and their products with (8) and (9) are called barycentric differential forms over T .
We simplify the notation whenever there is no danger of ambiguity: . With our choice of notation, the simplex T is always a superscript except for the barycentric monomials.

2.4.
Traces. Let T be an n-dimensional simplex and let F ⊆ T be an m-dimensional subsimplex of T . The trace from T to F is the mapping tr T,F : Λ k (T ) → Λ k (F ), which is the pullback along the inclusion ı(F, T ) : F → T introduced above. The trace commutes with the exterior derivative: tr T, The trace does not depend on the order of the vertices. However, taking into account the ordering of the vertices provides explicit formulas for traces of barycentric differential forms. Write [ı(F, T )] for the set of indices of those vertices of T that are also vertices of F .
, and in that case tr T,F λ T i = λ F j . Analogous observations hold for the exterior derivatives of the barycentric coordinates.
, then there exists a unique σ ∈ Σ(a : b, 0 : m) with ı(F, T ) • σ = σ, and then 2.5. Finite Element Spaces over Simplices. This subsection summarizes results about spanning sets and bases in finite element exterior calculus. Consider an n-dimensional simplex T , a polynomial degree r ∈ Z, and a form degree k ∈ Z. Suppose that either F = R or F = C. We introduce the sets of polynomial differential forms These are important spanning sets: the linear hulls i of the first two sets give rise to the standard finite element spaces of finite element exterior calculus with coefficients in the field F: Requiring the traces to vanish along the simplex boundary defines subspaces We know explicit spanning sets for these spaces as well. When r ≥ 1, then The first equation is also true when r = 0 and k < n, and the second equation is also true when r = 0; the vector spaces are trivial in those cases. The sets (12) are called the canonical spanning sets. ii The canonical spanning sets are generally not linearly independent and hence not bases. However, further constraining the indices in the canonical spanning sets produces the following bases (see [34]). When r ≥ 1, we define the sets of barycentric differential forms A particular feature of these bases and spanning sets are their inclusion relations. On the one hand, the bases are subsets of the spanning sets, . On the other hand, the generators for the spaces with boundary conditions are contained in the generators for the unconstrained spaces, i These linear hulls are trivial if r < 0 or k / ∈ [0 : n]. ii This is a minor abuse of terminology: the space FP 0 Λ n (T ) is one-dimensional but the corresponding set SP 0 Λ k (T ) is empty.
For any σ ∈ Σ(k, n) and ρ ∈ Σ 0 (k, n) we let the bubble functions λ T σ ∈ P k (T ) and λ T ρ ∈ P k+1 (T ) be defined by . Note how this also defines the bubble functions λ T σ c and λ T ρ c . With those bubble functions, we get yet another, explicit definition of the spanning sets (12) and bases (14) for the spaces with vanishing boundary traces: In the remainder of this document, we do not explicitly mention the field F when there is no danger of ambiguity.

The above bases and spanning sets for
and [9]. The above bases and spanning sets for P r Λ k (T ) andP r Λ k (T ) are discussed in [8], whereas [9] introduces different bases. This subsection summarizes [34,Section 4], which contributes alternative proofs.

Elements of Representation Theory
In this section we gather elements of the representation theory of finite groups. We keep this rather concise and refer to the literature [43,19,44,29,23] for thorough expositions on representation theory. We are particularly interested in the notions of irreducible representations, induced representations, and monomial representations. While the first two concepts are standard material in expositions on representation theory, the notion of monomial representation seems to have attracted much less attention yet.
Throughout this section we fix a finite group G. The binary operation of the group is written multiplicatively. We let e ∈ G denote the identity element of G and we let g −1 ∈ G be the inverse element of any g ∈ G. Furthermore, we fix F ∈ {R, C} in this section to be either the field of real numbers or the field of complex numbers. For any vector space V over F we write GL(V ) for its general linear group.
A representation of G is a group homomorphism r : G → GL(V ) from G into the general linear group of a vector space V . Necessarily, r(e) = Id V and for all g, h ∈ G we have r(gh) = r(g)r(h) and r(g) −1 = r(g −1 ). The dimension of r is defined as the dimension of V , and the representation r is called finite-dimensional if V is finite-dimensional. A representation is called faithful if it is a group monomorphism, that is, only the unit of the group is mapped to the identity.
We call two representations r : G → GL(V ) and s : In many circumstances, we are only interested in features of representations up to equivalence.
The most important example of a group in this article is the group Perm(a : b) of permutations of the set [a : b] for some a, b ∈ Z. The binary operation of the group is the composition. We also recall the cycle notation: when and leaves all other members of [a : b] invariant.

Example 3.2.
Let G be any group and let V be any vector space over the field F. Then the mapping r : G → GL(V ) that assumes the constant value Id V is a representation of G. This basic but important example is the so-called trivial representation of G. For another basic example, recall that every group G generates the vector space V = F G over F. The mapping r : G → GL(V ) such that r(g)h = gh for all g, h ∈ G is a representation of G.
3.1. Direct Sums, Subrepresentations, and Irreducible Representations. We want to compose new representations from old representations. One way of doing so is the direct sum. Let r : G → GL(V ) and s : G → GL(W ) be two representations of G. Their direct sum is another representation of G and is defined by The definition of the direct sum extends to the case of more than two direct summands in the obvious manner. We are interested in how to conversely decompose a representation into direct summands. To study that topic, we introduce further terminology.
Examples of r-invariant subspaces are V itself and the zero vector space. We call the representation r irreducible if the only r-invariant subspaces of V are the zero vector space and V itself; otherwise we call r reducible.
Suppose that W ⊆ V is an r-invariant subspace. Then there exists a representation r W : G → GL(W ) in the obvious way. We call r W a subrepresentation of r. The following result is known as Maschke's theorem [35].
and such that each r Vi is irreducible.
Proof. If r is irreducible, then there is nothing to show. Otherwise, there exists an r-invariant subspace W ⊂ V that is neither V itself nor the trivial subspace. We let P : V → W ⊆ V be any projection of V onto W . Since G is finite, we can define the linear mapping One verifies that S is again a projection onto W . Furthermore, we see that S(r(g)v) = r(g)S(v) for all g ∈ G and v ∈ V . So ker(S) is r-invariant. Since V = W ⊕ ker(S) by linear algebra, we decompose V into the direct sum of two non-trivial r-invariant subspaces. One then sees that r is the direct sum of the representations of G over these subspaces. Since V is finite-dimensional, an induction argument over the dimension of V shows the claim.

Restrictions and Induced Representations.
Let H ⊂ G be a subgroup of G. We recall that the cardinality of H divides the cardinality of G, and that the quotient |G|/|H| is called the index of H in G. Then we have a representation r H : H → V that is called the restriction of r to the subgroup H. Generally, we cannot recover the original representation from its restriction to a subgroup. However, there exists a canonical way of inducing a representation of a group from any given representation of one of its subgroups.
Suppose that we have a representation s : H → GL(W ) of the subgroup H over the vector space W . First, we let g 1 , g 2 , . . . , g M be any list of representatives of the left cosets of H in G, that is, where necessarily M = |G|/|H| is in the index of H in G. We recall that for every g ∈ G there exists a unique permutation τ g ∈ Perm(1 : M ) such that gg i ∈ g τ (i) H. More specifically, there exists a unique h g,i ∈ H such that gg i = g τ (i) h g,i . We now define the vector space and define a representation r : G → GL(V ) by setting componentwise In other words, We call this the induced representation. Conceptually, V consists of M copies of W , each associated to a coset representative g i , and the induced representation first applies the initial representation of H componentwise and then permutes the components.
We remark that the induced representation as defined above depends on the choice of representatives g 1 , g 2 , . . . , g M of the left cosets. Different sets of representatives lead to different induced representations, however, all those different representations are equivalent. Hence, technically, the literature defines induced representations only up to equivalence. We refer to [43,Chapter 12.5] for further background and details.

Monomial Representations and Invariant Sets.
A square matrix is called monomial, or generalized permutation matrix, if it is the product of a permutation matrix and an invertible diagonal matrix. Hence monomial matrices are the invertible matrices that have the non-zero pattern of a permutation matrix. A group representation r : G → GL(V ) is called monomial if there exists a basis of V with respect to which r(g) is a monomial matrix for each g ∈ G.
A representation of G is called induced monomial if it is induced by a one-dimensional representation of a subgroup H of G. It is easy to see that every induced monomial representation is monomial. We remark that many authors use the term monomial for what we call induced monomial. For irreducible representations, being monomial and being induced monomial are equivalent [19, Corollary 50.6].

Lemma 3.4.
If the representation r is irreducible and induced monomial, then r is monomial.
We now introduce the notion of invariance that is central to the following studies. To the author's best knowledge, the following is not standard terminology in the literature of representation theory. Suppose that Q ⊆ V is a set of M pairwise different vectors of V , We say that Q is F-invariant under r if for every g ∈ G there exists a permutation τ ∈ Perm(1 : M ) and a sequence of complex units χ 1 , . . . , χ M ∈ F such that We notice that any R-invariant subset of a real vector space gives rise to an R-invariant subset of the complexification of that vector space.

Notions of Invariance
In this section we connect the preceding elements of representation theory with finite element exterior calculus. We identify the pullback of barycentric differential forms along the affine automorphisms of a simplex as a representation of the symmetric group. Here and in all subsequent sections, we let F ∈ {R, C} be arbitrary unless mentioned otherwise.
We are particularly interested in the affine automorphisms of a simplex. Suppose that T is an n-simplex with vertices v 0 , . . . , v n , respectively. For any permutation π ∈ Perm(n) there exists a unique affine diffeomorphism S π : T → T such that We let Sym(T ) denote the symmetry group of T , which is the group of all affine automorphisms of T and whose members we call simplicial symmetries. We say iii that the permutation π induces the simplicial symmetry S π .
Since the simplicial symmetries are also diffeomorphisms, we can pullback differential forms along them. In the terminology of representation theory, we have representations that map permutations to the pullbacks along the corresponding simplicial symmetries. We briefly verify that this is indeed a representation of the group Perm(n). For π, µ ∈ Perm(n) we see Hence, S π•µ = S µ S π . Consequently, S * π•µ = (S µ S π ) * = S * π S * µ . So the mapping (16) does indeed define a group homomorphism and thus is a representation of Perm(n). Of course, this representation is not finite-dimensional.
We are interested in the subrepresentation of the permutation group over spaces of polynomial differential forms. We prepare this with several observations regarding the pullback operation on barycentric differential forms along S π . For any m, n ∈ Z, we write δ m,n for the Kronecker delta. For all i, j ∈ [0 : n] we observe that the pullback of the barycentric coordinates satisfies Since the pullback along affine mappings preserves affine functions, It follows that for any multiindex α ∈ A(n) we have For describing the pullback of barycentric differential forms along symmetry transformations, it suffices to consider basic alternators and Whitney forms. That is the content of the following two auxiliary lemmas. Let k ∈ [1 : n], σ ∈ Σ(k, n) and π ∈ Perm(n). Then where σ ∈ Σ(k, n) such that [ σ] = [πσ].
iii When Sym(T ) carries the composition as binary group operation, then the association π → Sπ is not a group homomorphism but a so-called antihomomorphism.
These observations suffice to completely describe the transformation of barycentric polynomial differential forms along affine diffeomorphisms. Evidently, the finite element spaces studied in this article are invariant under the representation of the permutation group. We have subrepresentations . Now we apply the notion of F-invariant set introduced in the preceding section. We say that a set Q ⊆  Proof. This follows from the definitions of these sets together with (18), (19), and (20).

Lemma 4.5.
Let r ∈ N. We have R-invariant bases Proof. Let r ≥ 1. In regard to 0-forms, definitions imply the identities . In regard to n-forms, one can show that Thus, λ α T dλ T σ ∈ BP r Λ n (T ). The desired R-invariance of those sets follows from these identities together with Lemma 4.3.
While all the canonical spanning sets are R-invariant, we have identified only a few R-invariant bases of finite element spaces. The remainder of the exposition will address the following question: under which circumstances do finite element spaces of differential forms have invariant bases in the sense of this subsection?

Invariant Bases of Lowest Polynomial Degree
We commence our study of invariant bases with the case of the constant differential k-forms over an nsimplex. Already the lowest-degree case exhibits non-trivial features. It serves as the base case for recursively constructing invariant bases in the last section. We utilize some advanced results in the representation theory of the symmetric group.
Lemma 5.1. Let T be an n-simplex and k ∈ N 0 . The representation of Perm(n) on FP 0 Λ k (T ) is irreducible. It is faithful for 0 < k < n.
Proof. If 0 < k < n, it is easily seen that the representation is faithful since only the identity element of Perm(n) acts as the identity on FP 0 Λ k (T ). That the representation is irreducible can be found in the literature [23, Proposition 3.12].
We first consider the tetrahedron. We build an R-invariant basis of RP 0 Λ 1 (T ), and then construct an R-invariant basis for RP 0 Λ 2 (T ) by taking the exterior power.

Lemma 5.2.
Let T be a 3-simplex. An R-invariant basis of RP 0 Λ 1 (T ) is In particular, this is also a C-invariant basis of CP 0 Λ 1 (T ).
Proof. An elementary calculation verifies that the set is a basis. The permutation group Perm(3) is generated by the three cycles (01), (02), and (03). Direct computation verifies that Hence, this set is R-invariant.
In particular, this is also a C-invariant basis of CP 0 Λ 2 (T ).
Proof. We immediately see that these three 2-forms are a basis of RP 0 Λ 2 (T ). Using the cycles (01), (02), and (03) as in the previous proof, direct computation shows Next we inspect the triangle, where the situation is more complicated: we need to consider not only real but also complex coefficients.
Proof. We easily check that the two vectors constitute a basis and that where we have used the cycles (01), (02) ∈ Perm(2). Since those are generators of Perm (2), it follows that {θ 0 , θ 1 } is a C-invariant basis of CP 0 Λ 1 (T ). Suppose that CP 0 Λ 1 (T ) has an R-invariant basis. Since our representation of Perm(2) over CP 0 Λ 1 (T ) is faithful by Lemma 5.1, it then follows that Perm(2) is isomorphic to a subgroup of the group of 2 × 2 signed permutation matrices. The latter group has order 8 whereas Perm(2) has order 6. This contradicts the well-known fact that the order of a group is divided by the orders of their subgroups. So CP 0 Λ 1 (T ) has no R-invariant basis.
Seemingly serendipitously, we present a C-invariant basis for the constant bivector fields over a 4-simplex.
Remark 5.6. Whereas Djoković and Malzan's results [21] include that a monomial representation of Perm(4) over CP 0 Λ 2 (T ) exists, they do not state an explicit basis and their argument is not immediately constructive. For that reason, we review how the aforementioned basis can be found.
Recall that Perm(4) is generated by the two cycles (01) and (01234). The 5-cycle (01234) is represented by a generalized permutation matrix of size 6 × 6, and so that matrix has the non-zero structure of a 6 × 6 permutation matrix of order 5. In particular, one of the C-invariant basis vectors must be invariant under the cyclic vertex permutation. We make the initial ansatz that the monomial matrices have coefficients in the quartic roots of unity. Via machine assisted brute-force search one finds 4 different vectors with that invariance property, up to multiplication by complex units.
With the additional ansatz that the 2-cycle (01) maps these invariant forms into the orbit of the aforementioned 5-cycle, one constructs five more vectors of the supposed basis. One then checks manually their linear independence and their C-invariance. Up to multiplication by complex units, this procedure only leaves the basis in Lemma 5.5 and its complex conjugate.
Remark 5.7. The following observations have been suggested by the anonymous referee and are included as a service for the reader. They shed new light onto the basis vectors above. For any 5-cycle g = (abcde) ∈ Perm(0 : 4), we let ζ g = ω g + iω g 2 , ω g = dλ ab + dλ bc + dλ cd + dλ de + dλ ea .
We immediately observe ω g −1 = −ω g . Together g 5 = e, one calculates So the 5-cycles generated by g induce the same ζ g up to quartic roots of unity. It is clear that relabeling the simplex vertices will send ζ g to ζ g ′ for some 5-cycle g ′ ∈ Perm(0 : 4).
Each 5-cycle g ∈ Perm(0 : 4) generates a cyclic subgroup of order 5. Since the entire group contains 4! different 5-cycles, we see that every 5-cycle must be belong to exactly one of six different cyclic subgroups. Upon choosing six 5-cycles g 1 , . . . , g 6 that generate the six different subgroups, the corresponding forms ζ g1 , . . . , ζ g6 are invariant up to complex units. We have already pointed out Djoković and Malzan's contribution [21] on monomial representations of the symmetric group. The invariant bases constructed in this section concretize their results. Apart from the constant scalar and volume forms, for which R-invariant bases are obvious, the bases found above already are exhaustive examples: no other spaces of constant differential forms over simplices of any dimension allows for C-invariant bases. That is the content of the following result. Proof. We recall that the representations of Perm(n) over P 0 Λ k (T ) are irreducible. Djoković and Malzan have shown [21,Theorem 1] that the only induced monomial irreducible representation of the group Perm(n) over spaces of constant differential forms are the trivial and the alternating representations, which corresponds to the group action on the space of constant functions and constant volume forms, the irreducible representations of Perm(2) and Perm (3), and an irreducible representation of Perm(4) on the space P 0 Λ 2 (T ) for any 4-dimensional simplex T . Thus Lemma 4.5, and Lemmas 5.2, 5.3, 5.4, and 5.5 cover the irreducible representations of symmetric groups over constant differential forms. iv All other irreducible representations iv The group Perm(0 : 3) also has a two-dimensional induced monomial irreducible (and hence monomial) representation, but this is of no interest in our applications.
of Perm(n) are not induced monomial. Since induced monomial irreducible representations are monomial, the theorem follows.

Canonical Isomorphisms
In this section we review the interaction of simplicial symmetries with the canonical isomorphisms in finite element exterior calculus. We show that the isomorphisms preserve F-invariance of sets. These isomorphisms were discussed in [8] and also [15]; we follow the discussion in [34], where it is shown that these isomorphisms can be described in terms of the canonical spanning sets. In that sense, the isomorphisms are natural for finite element exterior calculus.
Let k, r ∈ N 0 with r ≥ 0. Recall the canonical isomorphisms These are uniquely defined by the identities Note that these two identities prescribe the values of I k,r and J k,r over the canonical spanning sets, which are not necessarily linearly independent. However, one can show that these definitions nevertheless yield well-defined F-linear mappings [34].
Remark 6.1. The seminal idea of these isomorphisms is mapping between finite element spaces without and with boundary conditions via multiplication by monomial "bubble" functions. For example, in the case k = n, we have J n,r (f vol T ) = λ 0 λ 1 · · · λ n · f for all f ∈ P r Λ 0 (T ), where vol T denotes the volume form of T . The canonical isomorphisms generalize that idea.
The desired identity for the first canonical isomorphism follows.
Analogous calculations work for the other isomorphism. Let ρ ∈ Σ 0 (k, n) and ρ ∈ Σ 0 (k, n) satisfy [ ρ] = [πρ]. Let α ∈ A(r, n) and α = απ −1 . Then . Finally, we observe As a direct consequence of this theorem, the canonical isomorphisms and their inverses map F-invariant sets onto F-invariant sets. We will use the following important corollary for constructing F-invariant bases.

Corollary 6.3.
Let T be an n-simplex, and k, r ∈ N 0 with r ≥ 0. Then:

Traces and Extension Operators
In this section we study the relation of simplicial symmetries with traces, extension operators, and geometric decompositions of bases. The traces of F-invariant sets are F-invariant again. Conversely, we discuss extension operators that preserve F-invariant sets. An important result is that an F-invariant geometrically decomposed basis exists if and only if such bases exist for each component in the geometric decomposition.
We first prove that taking traces preserves F-invariance. Since Q is F-invariant, there exist units χ 1 , . . . , χ M ∈ F and a permutation τ ∈ Perm(1 : M ) such that There exists S F ∈ Sym(F ) which reorders the vertices of F in the same way as S does. We observe which had to be shown.
The idea of geometrically decomposed bases is central to finite element exterior calculus. Usually, geometrically decomposed bases are constructed explicitly via specific extension operators [9,34]. As a preparation, we introduce geometric decompositions on a slightly more abstract level where, importantly, we already study F-invariant sets.
Let T be an n-simplex and let k, r ∈ N 0 with r > 0. Suppose that QP r Λ k (T ) is a basis of P r Λ k (T ). We call such a basis geometrically decomposed if it is the disjoint union where F ranges over all the subsimplices of T , and where Q F satisfies, on the one hand, that the trace from T to F maps Q F bijectively onto a basis ofP r Λ k (F ), and on the other hand, that tr T,G Q F = {0} whenever G is a subsimplex of T not containing F . We define geometrically decomposed bases of P − r Λ k (T ) completely analogously.

Example 7.2.
As we shall discuss in more details below, our notion of geometric decomposition is only a minor generalization of earlier decompositions in the literature [8,9]. The bases BP r Λ k (T ) and BP − r Λ k (T ) are geometrically decomposed. Notably, not all of them are F-invariant. For further illustration, suppose that T is a triangle. The barycentric coordinates λ T 0 , λ T 1 , λ T 2 are a geometrically decomposed basis of P 1 Λ 0 (T ), whereas the basis is not geometrically decomposed. Both bases, however, are F-invariant.
Let T be an n-simplex and let k, r ∈ N 0 with r > 0. Let QP r Λ k (T ) be a basis of P r Λ k (T ) with geometric decomposition (26). Then Q T is a basis ofP r Λ k (T ). For any subsimplex G of T , a geometrically decomposed basis of P r Λ k (G) is given by Proof. Suppose that QP r Λ k (T ) is a geometrically decomposed basis of P r Λ k (T ). That Q T is a basis of P r Λ k (T ) follows from definitions. We show that QP r Λ k (G) is a basis of P r Λ k (G) when G is any subsimplex of T .
By assumption, tr T,G Q F = {0} if F is not a subsimplex of G. If instead F ⊆ G, then by assumption, tr T,F : Q F → tr T,F Q F is a bijection and its image is basis ofP r Λ k (F ). Because tr T,F Q F = tr G,F tr T,G Q F , the trace tr G,F : tr T,G Q F → tr T,F Q F is a bijection too.
That QP r Λ k (G) spans P r Λ k (G) is easily seen: To show that (27) defines a linearly independent set, let ω ∈ P r Λ k (G) be the sum of ω F ∈ span tr T,G Q F , F ⊆ G, not all zero. Then there exists F of minimal dimension with ω F = 0, and thus tr G,F ω = tr G,F ω F = 0. In particular, ω = 0. Lastly, that QP r Λ k (F ) is geometrically decomposed follows from the observations above. Suppose that QP r Λ k (T ) is F-invariant. By Lemma 7.1, the set of traces tr T,G QP r Λ k (T ) is F-invariant, and therefore its subset of non-zero traces is Theorem 7.4. Let T be an n-simplex and let k, r ∈ N 0 with r > 0. Let QP − r Λ k (T ) be a basis of P − r Λ k (T ) with geometric decomposition analogous to (26). Then Q T is a basis ofP − r Λ k (T ). For any subsimplex G of T , a geometrically decomposed basis of P − r Λ k (G) is given by Proof. This is completely analogous to the proof of Theorem 7.3.
Up to now, we have studied properties of any geometrically decomposed basis and how this definition interacts with our notion of invariance. Most importantly, invariant geometrically decomposed bases give rise to invariant decomposed bases for certain subspaces and trace spaces. Shifting our focus to extension operators, more specific statements are possible.
Extension operators that facilitate geometric decompositions are widely used in finite element exterior calculus [8,9,34]. For our purpose, we utilize the extension operators given by Arnold, Falk, and Winther [9]. Their extension operators are described over spanning sets, not bases, but this still yields well-defined linear mappings.
Let T be an n-dimensional simplex and let F ⊆ T be an m-dimensional subsimplex, and let k, r ∈ N 0 with r > 0. The extension operators for the P − r Λ k -family of spaces, are uniquely defined by setting for all ρ ∈ Σ 0 (k, m) and α ∈ A(r − 1, m), whereρ = ı(F, T ) • ρ ∈ Σ 0 (k, n), and whereα ∈ A(r − 1, n) is uniquely defined by requiringα • ı(F, T ) = α; in particular, α is zero outside of [ı(F, T )]. This prescribes the extension operator over the spanning set SP − r Λ k (F ) of the spaceP − r Λ k (F ), and one can show [9, Section 7] that this defines a linear operator.
As described in [9,Section 8], the extension operators are well-defined by setting for all σ ∈ Σ(1 : k, 0 : m) and α ∈ A(r, m), whereσ = ı(F, T ) • σ ∈ Σ(k, n), and whereα ∈ A(r, n) is uniquely defined by requiringα • ı(F, T ) = α; analogously to above, α is zero outside of [ı(F, T )]. This prescribes the extension operator over the spanning set SP r Λ k (F ) of the spaceP r Λ k (F ), and it follows from [9, Section 8] that this defines a linear operator. These operators are called extension operators because they are right-inverses of the trace, Moreover, whenever G is another subsimplex of T , then F ⊆ G implies whereas F G implies tr T,G ext k,r,− F,T ω = 0, ω ∈P − r Λ k (F ), tr T,G ext k,r F,T ω = 0, ω ∈P r Λ k (F ).
We refer to prior publications [9] for detailed discussion of these extension operators. The central result is the following decomposition.
Let T be an n-simplex and let k, r ∈ N 0 with r > 0. Then Remark 7.6. The geometric decomposition in Theorem 7.5 is fundamental to finite element theory and its importance can hardly be overstated: it is the geometric decomposition which enables the construction of localized bases. We refer to the literature [9] for further background.
The above results on extension operators are known. Next we study how these extension operators interact with simplicial symmetries.
We introduce additional notation. Let F ⊆ T be a subsimplex of a simplex T and let S ∈ Sym(T ). Then SF is a subsimplex of T of the same dimension as F , and we have affine diffeomorphisms Theorem 7.7. Let k, r ∈ N 0 with r > 0. Let T be a simplex of dimension n and F ⊆ T be a subsimplex of dimension m. If S ∈ Sym(T ), then ext k,r F,T S * |F = S * ext k,r SF,T , ext k,r,− F,T S * |F = S * ext k,r,− SF,T .
Proof. It suffices to prove both identities over the canonical spanning sets. For every S ∈ Sym(T ) and every subsimplex F of T , we can decompose S = S 1 S 2 for some S 1 , S 2 ∈ Sym(T ) where S 1|F : F → SF preserves the order of vertices and S 2 (F ) = F . It suffices to consider simplicial symmetries S belonging to one of the two special cases.
We now work along the following idea: if a basis allows for a geometric decomposition corresponding to Theorem 7.5, then this basis is F-invariant under the provision that the components in the geometric decomposition satisfy certain invariance properties. This is formalized in the following two theorems, which strengthen Theorems 7.3 and 7.4. An important consequence is this: F-invariant bases for the components in that geometric decomposition yield F-invariant bases for the entire finite element space over the simplex.
Theorem 7.8. Let k, r ∈ N 0 with r > 0 and let T be an n-simplex. Assume that QP r Λ k (F ) is a basis forP r Λ k (F ) for each subsimplex F ⊆ T , and define Then QP r Λ k (T ) is a basis of P r Λ k (T ). The following statements are equivalent: • For each subsimplex F of T the set QP r Λ k (F ) is F-invariant and for each S ∈ Sym(T ) that preserves the relative order of vertices of F we have Proof. For any subsimplex F ⊆ T , since QP r Λ k (F ) is a basis forP r Λ k (F ) we see that ext k,r F,T QP r Λ k (F ) is a basis for ext k,r F,TP r Λ k (F ), via Theorem 7.7. By Theorem 7.5 then, it is clear that QP r Λ k (T ) is a basis of P r Λ k (T ).
Assume that QP r Λ k (T ) is F-invariant. Let F ⊆ T be a subsimplex and let S ∈ Sym(F ). There exists S ∈ Sym(T ) such that S = S |F . The pullback along S preserves the space ext k,r F,TP r Λ k (F ) since Theorem 7.7 implies S * ext k,r F,TP r Λ k (F ) = ext k,r F,T S * P r Λ k (F ) = ext k,r F,TP r Λ k (F ).
Because QP r Λ k (T ) is F-invariant, for every ω 1 ∈ QP r Λ k (F ) there exists ω 2 ∈ QP r Λ k (F ) and a complex unit χ ∈ F such that S * ext k,r F,T ω 1 = χ ext k,r F,T ω 2 . Using Theorem 7.7 again, we note that S * ω 1 = tr T,F ext k,r F,T S * ω 1 = tr T,F S * ext k,r F,T ω 1 = χ tr T,F ext k,r F,T ω 2 = χω 2 . Hence, by definition, Let F ⊆ T be a subsimplex and let S ∈ Sym(T ) be such that the mappings preserve the relative order of the vertices. By Theorem 7.7, we have S * ext k,r SF,TP r Λ k (SF ) = ext k,r F,T S * |FP r Λ k (SF ) = ext k,r F,TP r Λ k (F ). The same argument can be applied to the inverse of S; it follows that we have an isomorphism S * : ext k,r SF,TP r Λ k (SF ) → ext k,r F,TP r Λ k (F ). As QP r Λ k (T ) is F-invariant, S * maps ext k,r SF,T QP r Λ k (SF ) bijectively onto ext k,r F,T QP r Λ k (F ) up to multiplication by units of F. But since S preserves the relative ordering of the vertices of F , a direct calculation shows that those units must equal one. Hence we have a bijection S * : ext k,r SF,T QP r Λ k (SF ) → ext k,r F,T QP r Λ k (F ). Thus, we have shown that the first statement of the theorem implies the second statement.
It remains to show the converse implication, so let us assume that the second statement is true. Let S ∈ Sym(T ). There exist S 1 , S 2 ∈ Sym(T ) such that S = S 1 S 2 , we have S 1 (F ) = SF and S 2 (F ) = F , and S 1|F : F → SF preserves the order of vertices.
Let ω ∈ QP r Λ k (T ). There exists a subsimplex F of T and ω 0 ∈ QP r Λ k (SF ) such that ω = ext k,r SF,T ω 0 . Note that by assumption, we have a bijection S * 1 : ext k,r SF,T QP r Λ k (SF ) → ext k,r F,T QP r Λ k (F ). Hence, there exists ω 1 ∈ QP r Λ k (F ) such that S * 1 ext k,r SF,T ω 0 = ext k,r F,T ω 1 . Furthermore, since QP r Λ k (F ) is assumed to be F-invariant, there exist a complex unit χ ∈ F and ω 2 ∈ QP r Λ k (F ) such that S * 2|F ω 1 = χω 2 . Thus S * ω = S * 2 S * 1 ext k,r SF,T ω 0 = S * 2 ext k,r F,T ω 1 = ext k,r F,T S * 2|F ω 1 = χ ext k,r F,T ω 2 . As a consequence, QP r Λ k (T ) is F-invariant. Theorem 7.9. Let k, r ∈ N 0 with r > 0 and let T be an n-simplex. Assume that QP − r Λ k (F ) is a basis forP − r Λ k (F ) for each subsimplex F ⊆ T , and define Then QP − r Λ k (T ) is a basis of P − r Λ k (T ). The following statements are equivalent: is F-invariant and for each S ∈ Sym(T ) that preserves the relative order of vertices of F we have Proof. This is proven completely analogously to the preceding theorem.
A geometrically decomposed basis enjoys F-invariance if each component in the geometric decomposition is already F-invariant and if, additionally, the components associated to subsimplices of the same dimension are "reindexings" of each other. That second additional condition is a mere technicality: if we know an Finvariant basis for the spaces of vanishing trace over, say, any m-dimensional reference simplex, then pullback along vertex-order preserving affine diffeomorphisms immediately defines such components associated to every m-dimensional subsimplex. Hence the only question of technical interest remaining for constructing geometrically decomposed F-invariant bases is whether F-invariant bases are known for the corresponding spaces with vanishing boundary traces on all subsimplices.

Recursive Basis Construction
This final section describes the recursive construction of geometrically decomposed bases for simplicial higher-degree finite element spaces in finite element exterior calculus. We combine the results of the previous sections.
We commence the construction as follows. First, for every simplex T we fix a basis AP 0 Λ k (T ) for the lowest-degree space P 0 Λ k (T ). In principle, any arbitrary choice of bases can serve as the base case in our recursive construction. But some specific choices of bases, given further below, lead to F-invariant higherdegree bases.
Moreover, recall that the empty set is a basis for the trivial vector space. To simplify some technical arguments, we therefore fix the empty bases whenever the corresponding vector space is trivial. Specifically, we let AP r Λ k (T ) = ∅ when r < 0 and AP − r Λ k (T ) = ∅ when r ≤ 0. Similarly, we set AP − r Λ k (T ) = ∅ when r < n − k + 1, and we set AP r Λ k (T ) = ∅ either when k < n and r < n − k + 1 or when k = n and r < 0.
Second, recall the canonical isomorphisms of Section 6, . These isomorphisms are defined for r ≥ 0 and map bases to bases. Shifting indices, we construct bases for the finite element spaces with homogeneous boundary traces as follows. When r ≥ n − k, we set , provided that the bases AP r−n+k Λ n−k (T ) and AP − r−n+k+1 Λ n−k (T ) have already been constructed. As seen in Corollary 6.3, the bases AP − r+1 Λ k (T ) and AP r+1 Λ k (T ) are F-invariant if and only if the bases AP r−n+k Λ n−k (T ) and AP − r−n+k+1 Λ n−k (T ) are F-invariant, respectively.
Third, taking into account the geometric decompositions we define bases provided that the corresponding bases for the finite element spaces with boundary conditions have already been constructed.
The basis AP r Λ k (T ) is F-invariant if the bases AP r Λ k (F ) for every F satisfy the conditions of Theorem 7.8. Analogously, the basis AP − r Λ k (T ) is F-invariant if the bases AP − r Λ k (F ) for every F satisfy the conditions of Theorem 7.9. In both cases, F-invariant bases on the lower-dimensional subsimplices are easily obtained if we know F-invariant bases on any simplices of the corresponding dimensions.
It is now clear how to recursively construct bases for finite element spaces of arbitrary polynomial degree: starting from any given bases for constant differential forms, we construct bases of higher polynomial degree, using the canonical isomorphisms or the geometric decomposition. In particular, if the initial zero-degree finite element bases are F-invariant, then the recursively constructed higher-degree finite element bases will be F-invariant as well.
We develop this idea when the simplex T has practically relevant low dimension. As explained above, our construction of higher-degree F-invariant bases relies on F-invariant bases for the zero-degree spaces, which constitute the base case of the recursion. For the remainder of this section, we therefore make the following additional assumptions: • On any simplex T , the basis AP 0 Λ 0 (T ) consists of the constant function with pointwise value 1.
• On any n-dimensional simplex T , the bases AP 0 Λ n (T ) and AP 0 Λ n (T ) consist of the volume form vol T . • If T is a triangle, then AP 0 Λ 1 (T ) is the C-invariant basis of Lemma 5.4.
• If T is a tetrahedron, then AP 0 Λ 1 (T ) and AP 0 Λ 2 (T ) are the R-invariant bases of Lemmas 5.2 and 5.3.
• If T is a 4-simplex, then AP 0 Λ 2 (T ) is the C-invariant basis of Lemma 5.5.
This stipulation leads to the following results. We begin with finite element bases for higher-degree scalar and volume forms to illustrate the basic ideas.
Let T be an n-dimensional simplex, and let r ∈ N 0 . Then the bases Proof. We perform nested induction over the dimension n and the polynomial degree r. The base case are simplices of dimension n = 0, that is, vertices. So suppose that n = 0. The statement holds when r = 0, where the bases AP 0 Λ 0 (T ) and AP 0 Λ 0 (T ) are R-invariant, and the sets AP − 0 Λ 0 (T ) and AP − 0 Λ 0 (T ) are empty. Suppose the statement is true for some polynomial degree r ≥ 0. Note that AP r Λ 0 (T ) cannot be empty. The first canonical isomorphism maps AP r Λ 0 (T ) to AP − r+1 Λ 0 (T ), and the geometric decomposition trivially leads to AP − r+1 Λ 0 (T ). The second canonical isomorphism maps AP − r+1 Λ 0 (T ) to AP r+1 Λ 0 (T ), and the geometric decomposition now leads to AP r+1 Λ 0 (T ). By construction, the statement thus holds for the polynomial degree r + 1. The principle of induction implies the statement for any polynomial degree when n = 0.
Suppose that T is an arbitrary but fixed simplex of dimension n and that the statement is true for all polynomial degrees on simplices up to dimension n − 1. We then use induction over the polynomial degree r.
We first consider the case r = 0. We have R-invariant bases AP 0 Λ 0 (T ) and AP 0 Λ n (T ). The set AP 0 Λ 0 (T ) is empty and the basis AP 0 Λ n (T ) is R-invariant. The sets AP − 0 Λ 0 (T ), AP − 0 Λ n (T ), AP − 0 Λ n (T ) and AP − 0 Λ n (T ) are empty. Next, we assume the statement holds up to polynomial degree r−1 ≥ 0 and prove it for degree r. Using that r ≥ 1, the first canonical isomorphism maps AP r−1 Λ 0 (T ) onto AP − r Λ n (T ). We immediately get the basis AP − r Λ n (T ) from the geometric decomposition. When r ≤ n, then AP − r Λ 0 (T ) is empty, and otherwise, the first canonical isomorphism maps AP r−n−1 Λ n (T ) onto AP − r Λ 0 (T ). The geometric decomposition provides the basis AP − r Λ 0 (T ). Using that r ≥ 1, the second canonical isomorphism maps AP − r Λ 0 (T ) onto AP r Λ n (T ). We immediately get the basis AP r Λ n (T ). When r ≤ n, then AP − r Λ 0 (T ) is empty, and otherwise, the second canonical isomorphism maps AP − r−n Λ n (T ) onto AP r Λ 0 (T ). The geometric decomposition provides the basis AP r Λ 0 (T ).
We can now apply the principle of induction over the polynomial degree r. Lastly, we have completed the induction over the dimension of the simplex. We conclude that the statement holds for all polynomial degrees r and over simplices T of any dimension n.
Similar techniques lead to the main results of this section, which now follow.
Proof. This follows the same line as the proofs of Theorem 8.2 and Theorem 8.5. The base case is addressed by Lemma 5.5.
are associated with the tetrahedron itself. We study the elementwise mass matrix in the special case where T is a regular tetrahedron. In contrast to scalar forms, some non-trivial orthogonality relations arise. One checks that the bases AP 0 Λ 1 (T ) and AP 0 Λ 2 (T ) are each orthogonal with respect to the respective standard products. One verifies the same for each of the bases AP − 2 Λ 2 (T ) and AP − 3 Λ 1 (T ), obtained along the respective canonical isomorphisms. Similarly, suppose that F is a regular triangle, for example, a face of T . Calculations show that the basis AP 0 Λ 1 (F ) is orthogonal with respect to the standard products, as is AP − 2 Λ 1 (F ) and the extension ext 1,2,− F,T AP − 2 Λ 1 (F ). While equilateral triangles and regular tetrahedra are obviously idealized special cases, these exemplary orthogonality relations suggest further computational studies.
Remark 8.9. Our search for invariant bases has led to positive results: Theorem 8.2, Theorem 8.5, and Theorem 8.7 explicitly construct C-invariant geometrically decomposed bases. We have pointed out conditions on the polynomial degree for the existence of bases that are R-invariant, that is, invariant under permutation of indices up to sign change. These conditions on R-invariance are sufficient and we conjecture that they are necessary as well.
If instead one of the finite element spaces above has an R-invariant basis for a polynomial degree r not listed above, then that basis must satisfy constraints related to geometric decompositions in the sense of Section 7. Because of Theorem 7.3 and 7.4, any geometrically decomposed R-invariant basis induces Rinvariant bases on subsimplices and for finite element spaces with vanishing boundary traces, and we can then use a canonical isomorphism to get R-invariant bases on spaces of lower degree.
For example, Djoković and Malzan's results already rule out R-invariant bases for several spaces of constant k-forms. Going further, there is no R-invariant basis forP − 2 Λ 1 (F ) or geometrically decomposed R-invariant basis for P − 2 Λ 1 (F ) over a triangle F . A fortiori, P − 2 Λ 1 (T ) over a tetrahedron T has no geometrically decomposed R-invariant basis.
Remark 8.10. Throughout this section, we have emphasized the "bottom-up" recursive construction of higher-degree bases. Conversely, given a basis for a higher-degree finite element space, we can follow the recursion "top-down", under the condition that the geometric decomposition can be used at every step. The latter condition is not necessarily true for every basis. In any case, no further recursion step is possible whenever the recursion leads to zeroth-degree finite element spaces: there, no geometric decomposition exists.