Preserving Interpolation Erick Rodriguez Bazan, Evelyne Hubert

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Erick Rodriguez Bazan, Evelyne Hubert. Symmetry Preserving Interpolation. ISSAC 2019 - International Symposium on Symbolic and Algebraic Computation, Jul 2019, Beijing, China. ￿10.1145/3326229.3326247￿. ￿hal-01994016￿

HAL Id: hal-01994016 https://hal.archives-ouvertes.fr/hal-01994016 Submitted on 25 Jan 2019

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Erick Rodriguez Bazan Evelyne Hubert Université Côte d’Azur, France Université Côte d’Azur, France Inria Méditerranée, France Inria Méditerranée, France [email protected] [email protected]

ABSTRACT general concept. An interpolation space for a of linear forms is The article addresses multivariate interpolation in the presence of a subspace of the polynomial ring that has a unique interpolant for symmetry. Interpolation is a prime tool in algebraic computation each instantiated interpolation problem. We show that the unique while symmetry is a qualitative feature that can be more relevant interpolants automatically inherit the symmetry of the problem to a mathematical model than the numerical accuracy of the pa- when the interpolation space is (Section 3). rameters. The article shows how to exactly preserve symmetry A canonical interpolation space, the least interpolation space, was in multivariate interpolation while exploiting it to alleviate the introduced in [3–5]. We shall review that it is invariant as soon as computational cost. We revisit minimal degree and least interpo- the space of linear forms is. In floating point arithmetics though, lation with symmetry adapted bases, rather than monomial bases. the computed interpolation space might fail to be exactly invariant. This allows to construct bases of invariant interpolation spaces in Yet, in mathematical modeling, symmetry is often more relevant blocks, capturing the inherent redundancy in the computations. than numerical accuracy. We shall remedy this flaw and further We show that the so constructed symmetry adapted interpolation exploit symmetry to mitigate the cost and numerical sensitivity of bases alleviate the computational cost of any interpolation problem computing a minimal degree or least interpolation space. and automatically preserve any equivariance of this interpolation As other minimal degree interpolation spaces, the least inter- problem might have. polation space can be constructed by Gaussian elimination in a multivariate Vandermonde (or collocation) matrix. The columns of KEYWORDS the Vandermonde matrix are indexed by monomials. We show how any other graded basis of the polynomial ring can be used. In partic- Interpolation, Symmetry, , Action ular there is a two fold gain in using a symmetry adapted basis. On one hand, the computed interpolation space will be exactly invari- 1 INTRODUCTION ant independently of the accuracy of the data for the interpolation Preserving and exploiting symmetry in algebraic computations problem. On the other hand, the new Vandermonde matrix is block is a challenge that has been addressed within a few topics and, diagonal so that Gaussian elimination can be performed indepen- mostly, for specific groups of symmetry2 [ , 7, 8, 10, 11, 13–16, 18, dently on smaller size matrices, with better conditioning. Further 19, 22]. The present article addresses multivariate interpolation in computational savings result from identical blocks being repeated the presence of symmetry. Due to its relevance in approximation according to the degree of the related irreducible representations theory and geometrical modeling, interpolation is a prime topic in of the group. Symmetry adapted bases also plaid a prominent role algebraic computation. Among the several problems in multivariate in [2, 11, 19] where it allowed the block diagonalisation of a multi- interpolation [9, 17], we focus on the construction of a polynomial variate Hankel matrix. interpolation space for a given set of linear forms. Assuming the In Section 2 we define minimal degree and least interpolation space generated by the linear forms is invariant under a group space and review how to compute a basis of it with Gaussian elimi- action, we show how to, not only, preserve exactly the symmetry, nation. In Section 3 we make explicit how symmetry is expressed but also, exploit it so as to reduce the computational cost. and the main ingredient to preserve it. In Section 4 we review symmetry adapted bases and show how the Vandermonde matrix For a set ofr points ξ1,..., ξr inn-space, andr valuesη1,..., ηr the basic interpolation problem consists in finding a n-variate poly- becomes block diagonal in these. This is applied to provide an algorithm for the computation of invariant interpolation spaces nomial p such that p(ξi ) = ηi , for 1 ≤ i ≤ r. The evalu- in Section 5 together with a selection of relevant invariant and ations at the points ξi form a basic example of linear forms. The space they generate is invariant under a when the set equivariant interpolation problems. of points is a union of orbits of this group action. A first instance of symmetry is invariance. The above interpolation problem is in- variant if ηi = ηj whenever ξi and ξj belong to the same orbit. It is then natural to expect an invariant polynomial as interpolant. Yet, contrary to the univariate case, there is no unique interpolant of minimal degree and the symmetry of the interpolation problem 2 POLYNOMIAL INTERPOLATION may very well be violated (compare Figure 2 and 1). We review in this section the definitions and constructions of in- In this article we shall consider a general set of linear forms, in- terpolation spaces of minimal degree. By introducing general dual variant under a group action, and seek to compute interpolants that polynomial bases we generalize the construction of least interpola- respect the symmetry of the interpolation problem. We mentioned tion spaces. We shall then be in a position to work with adapted invariance as an instance of symmetry, but equivariance is the more bases to preserve and exploit symmetry. Pδ 2.1 Interpolation space WL . They index a maximal set of linearly independent columns Hereafter, K denotes either C or R. K[x] = K[x ,..., x ] denotes Pδ 1 n of WL . the ring of polynomials in the variables x1,..., xn with coefficients in K; K[x]≤δ and K[x]δ the K−vector spaces of polynomials of 2.3 Minimal degree degree at most δ and the space of homogeneous polynomials of It is desirable to build an interpolation space such that the degree degree δ respectively. of the interpolating polynomials be as small as possible. We shall The dual of K[x], the set of K−linear forms on K[x], is denoted by ∗ use the definition of minimal degree solution for an interpolation K[x] . A typical example of a on K[x] is the evaluation problem defined in [4, 5, 20]. e n ξ at a point ξ of K . It is defined by e Definition 2.1. An interpolation space P for Λ is of minimal de- ξ : K[x] → K p 7→ p(ξ ). gree if for any other interpolation space Q for Λ Other examples of linear forms on K[x] are given by compositions dim(Q ∩ K[x]≤δ ) ≤ dim(P ∩ K[x]≤δ ), ∀δ ∈ N. of evaluation and differentiation Λ : K[x] → K We say that a countable set of homogeneous polynomials P = 7→ Pr e ◦ ∂ , {p1,p2,...} is ordered by degree if i ≤ j implies that degpi ≤ degpj . p j=1 ξj qj ( )(p) n α ∂ ∂ with ξ ∈ K ,q ∈ K[x] and ∂ = α ... α . Proposition 2.2. Let L be a basis of Λ. Let Pδ , δ > 0, be a j j 1 ∂x n ∂x1 n Pδ n homogeneous basis of K[x]≤ ordered by degree, such that W has Let ξ1,..., ξr be a finite set of points in K . Lagrange interpola- δ L ∈ Pδ tion consists in finding, for any η1,... ηr K, a polynomial p such full row rank. Let j1,..., jr be the echelon sequence of WL obtained e (p) = η ≤ j ≤ r interpolation problem that ξj j , 1 . More generally an by Gauss elimination with partial pivoting. Then P := ⟨pj1 ,...,pjr ⟩ is a pair (Λ, ϕ) where Λ is a finite dimensional linear subspace of is a minimal degree interpolation space for Λ. K[x]∗ and ϕ : Λ −→ K is a K-. An interpolant, i.e., a , ... solution to the interpolation problem, is a polynomial p such that Proof. LetQ be another interpolation space for Λ. Letq1 q2 qm be a basis of Q ∩ K[x]≤d with d ≤ δ. Since Pδ is a homogeneous λ(p) = ϕ(λ) for any λ ∈ Λ. (1) basis of K[x]≤δ , any qi can be written as a linear combination of P elements of P ∩ K[x]≤ . Considering qi = ajipj we get that P δ d j An interpolation space for Λ is a polynomial subspace P of K[x] λ(qi ) = j aji λ(pj ) for any λ ∈ Λ. such that Equation (1) has a unique solution in P for any map ϕ. Let {p ,p ,... p } be the elements of P that form a basis of ji1 ji2 jin Pδ P ∩ K[x]≤d . Gauss elimination on WL ensures that λ(b) is a linear 2.2 Vandermonde matrix ∈ P ∩K ∈ combination of λ(pji ),... λ(pjin ) for any b δ [x]≤d and λ 1 Pn For P = {p1,p2,...,pm } and L = {λ1, λ2,..., λr } linearly inde- Λ. The latter implies that λ(q ) = c λ(p ) for 1 ≤ i ≤ m i k=1 ki jik pendent sets of K[x] and K[x]∗ respectively, we introduce the   and cki ∈ K. If m > n then the matrix C = cij 1≤i ≤m has linearly (generalized) Vandermonde matrix 1≤j ≤n independent columns, and therefore there exist d ,d ,... d ∈ K   1 2 m P Pm Pm ∈ WL := λi pj 1≤i ≤r . (2) such that i=1 di λ(qi ) = λ( i=1 diqi ) = 0 for any λ Λ which f g1≤j ≤m is a contradiction with the fact that Q is an interpolation space of As in the univariate case, the Vandermonde matrix appears naturally Λ. Then we can conclude that m ≤ n and P is a minimal degree interpolation space for Λ. □ in the interpolation problem. spanK(P) is an interpolation space P for spanK(L) if and only if WL is an invertible matrix. This leads to a straightforward approach to compute an interpolation space 2.4 Duality and apolar product for ⟨L⟩. Since the elements of L are linearly independents, there ∗ P K[x] can be identified with the ring of formal power series K[[∂]] δ P ∗ is δ > 0 such that WL has full row rank, where δ is a basis through the Φ : K[[∂]] −→ K[x] , where for p = of K[x] . For Lagrange interpolation δ ≤ |L|. Hence we can P α P α ≤δ α pα x ∈ K[x] and f = α ∈Nn fα ∂ ∈ K[[∂]] Pδ choose r linearly independent columns j1, j2,... jr of WL and the X ∂α p X P = (p ,... p ) Φ(f )(p) := fα (0) = α!fα pα . corresponding space spanK j1 jk is an interpolation ∂xα space for Λ. α ∈Nn α ∈N P e n δ For instance, the evaluation ξ at a point ξ ∈ K is represented by In order to select r linearly independent columns of WL we can P (ξ, ∂)k Pδ e(ξ, ∂) = , the power series expansion of the exponen- use any rank revealing decomposition of WL . Singular value de- k ∈N k! composition (SVD) and QR decomposition provide better numerical tial function with frequency ξ . The dual pairing accuracy but to obtain a minimal degree interpolation space we K[x]∗ × K[x] → K shall resort to Gauss elimination. It produces a LU factorization of (λ,p) → λ(p) Pδ WL where L is an invertible matrix and U = uij 1≤i ≤r is in row K ∈ K ∂ ∈ ≤j ≤m brings the apolar product on [x] by associating p [x] to p( ) f g1 P α P α echelon form. This that there exists an increasing sequence K[[∂]]. For p = α pα x and q = α qα x the apolar product

between p and q is given by p,q := p(∂)q = P α!p q ∈ K. j1,..., jr with ji ≥ i, such that uiji is the first non-zero entry in α α α n n D t E the i−th row of U. We call j1,..., jr the echelon index sequence of Note that for a linear map a : K → K , p,q ◦ a = p ◦ a ,q . 2 For a set of linearly independent homogeneous polynomials P The we shall deal with are given by the linear group we define the dual set P† to be a set of homogeneous polynomials action of a finite group G on Kn. It is thus given by a representation D E † ϑ of G on Kn. It induces a representation ρ of G on K[x] given by such that pi ,pj = δij . For instance the dual basis of the monomial α 1 α ∗ basis {x } n is { x } n . Thus any linear form λ ∈ K[x] α ∈N α! α ∈N −1 . P 1 ρ(д)p(x) = p(ϑ (д )x) n α α ∈ can be written as λ = α ∈N α! λ(x )∂ K[[∂]]. More generally, P † any linear form on ⟨P⟩ can be written as λ = p ∈P λ(p)p (∂). K[x]δ is invariant under ρ. It also induces a linear representation on the space of linear forms, the dual representation of ρ : 2.5 Least interpolation space ∗ = −1 , ∈ K ∈ K ∗. For a space of linear forms Λ ⊂ K[x]∗, a canonical interpolation ρ (λ)(p) λ(ρ(д )p) p [x] and λ [x] space Λ is introduced in [5]. It has a desirable set of properties. ↓ We shall deal with an invariant subspace Λ of K[x]∗. Hence the An algorithm to build a basis of Λ based on Gauss elimination ↓ restriction of ρ∗ to Λ is a linear representation of G in Λ. on the Vandermonde matrix is presented in [4]. In this algorithm the authors consider the Vandermonde matrix associated to the monomial basis of K[x]. The notion of dual bases introduced above, 3.1 Invariance Definition 3.1. Let Λ be a space of linear forms and ϕ : Λ −→ K allows to extend the algorithm to any graded basis of K[x].   The initial term of a power series λ ∈ K[[∂]], denoted by λ↓ ∈ a linear map. The pair Λ, ϕ defines an invariant interpolation K[x] in [3–5], is the unique homogeneous polynomial for which problem if − ∂ λ λ↓( ) vanishes to highest possible order at the origin. Given a (1) Λ is closed under the action of G. linear space of linear forms Λ, we define Λ↓ as the of all (2) ϕ(ρ∗(д)(λ)) = ϕ(λ) for any д ∈ G and λ ∈ Λ. λ↓ with λ ∈ Λ. [5, Proposition 2.10] shows that dim Λ = dim Λ↓. An invariant Lagrange interpolation problem can be seen as inter- Proposition 2.3. Let P = {p1,p2,...} be a homogeneous basis polation at union of orbits of points with fixed values on their orbits, of K[x] ordered by degree and L = {λ1,..., λr } be a basis of Λ. Let i.e., given ξ ,..., ξ with orbits O ,..., O and η ,...,η ∈ Kn, LU = WP be the factorization of WP provided by Gauss elimination 1 m 1 m 1 m L L an interpolant p ∈ K[x] is to satisfy p ◦ ϑ (д) (ξ ) = η for any with partial pivoting with {j , j ,..., j } as echelon index sequence. k k 1 2 r д ∈ G. It is natural to expect that an appropriate interpolant p be If U = (u ) consider, for 1 ≤ ℓ ≤ r, ij invariant. Yet, not all minimal degree interpolants are invariant. X = † hℓ uℓk pk (3) Example 3.2. The dihedral group Dm is the group of order 2m deg(p )=deg(pj ) k ℓ that leaves invariant the regular m-gon. It thus has a representation P† { † † } P in R2 given by the matrices where = p1,....pj ,... is the dual basis of with respect to the apolar product. Then H = {h1,... hr } is a basis for Λ↓.     cos ⌊ k ⌋ 2π − sin ⌊ k ⌋ 2π !k X 2 m 2 m 1 0 −1 † ϑk = , 0 ≤ k ≤ 2m−1. Proof. Let L = (aij ) and consider ςℓ = uℓ p (∂). Since  k 2π   k 2π  0 −1 j j  sin ⌊ ⌋ cos ⌊ ⌋  j ∈N * 2 m 2 m +  . /  (4) Xr X  Xr    † Consider, Ξ ⊂ R2 a set of 1+3×5 points- illustrated on Figure 1. They uℓj = ali λi (pj ), then ςℓ = ali λi (pj ) pj (∂) O O O O e | ∈ i=1 j ∈N i=1 form four orbits 1, 2, 3, 4 of D5 so that Λ := span( ξi ξi Ξ) r r is invariant. An invariant interpolation problem is given by the pair X X † X e e = a λ (p )p (∂) = a λ ∈ Λ. (Λ, ϕ) where ϕ is defined by ϕ( ξ ) = 0.1 if ξ ∈ O1, ϕ( ξ ) = 0 if li i j j li i e i=1 j ∈N i=1 ξ ∈ O2 ∪ O4, and ϕ( ξ ) = −0.5 if ξ ∈ O3. We show in Figure 1 the graph of the expected interpolant, but in Figure 2 we present the Notice that h = ς and therefore h ∈ Λ for 1 ≤ ℓ ≤ r. ℓ ℓ ↓ ℓ ↓ graph of an interpolant of minimal degree. The ji are strictly increasing so that {h1,h2,...,hr } are linearly independent. Since dim(Λ) = dim(Λ↓) = r we conclude that H is a Proposition 3.3. Let (Λ, ϕ) be an invariant interpolation prob- basis of Λ↓. □ lem. Let P be an invariant interpolation space and let p ∈ K[x] be the solution of (Λ, ϕ) in P. Then p ∈ K[x]G , the ring of invariant polynomials. 3 SYMMETRY We define the concepts of invariant interpolation problem (IIP) and Proof. For any λ ∈ Λ and д ∈ G we have that λ(p) = ϕ(λ) equivariant interpolation problem (EIP). These interpolation prob- and ρ∗(д)(λ)p = ϕ(ρ∗(д)(λ)). Since ϕ is G−invariant, we get that lems have a structure that we want to be preserved by the inter- λ(ρ(д−1)p) = ρ∗(д)(λ)p = ϕ(ρ∗(д)(λ)) = λ(p) for any λ ∈ Λ. polant. We show that this is automatically achieved when choosing The latter implies that ρ(д−1)p − p ∈ Ker Λ. As P is closed under the interpolant in an invariant interpolation space. Then the solu- the action of ρ, ρ(д−1)p − p ∈ KerΛ T P. Then as (Λ, P) is an tion of an IIP is an invariant polynomial and the solution of an EIP interpolation space Ker Λ T P = ∅ and we conclude that ρ(д−1)p − is an equivariant polynomial map. In Section 5 we show that the p = 0 for any д ∈ G, i.e., p ∈ K[x]G . □ least interpolation space is invariant and how to better compute an invariant interpolation space of minimal degree. 3 Proof. For any λ ∈ Λ we have the following ∗ ∗ #Nodes A node per orbit ρ (д)(λ)f = ϕ(ρ (д)λ) = θ (д)ϕ(λ) = θ (д)λ(f ) = λ(θ (д)f ). (5) O 1 ξ = (0, 0) 1 1 m m O2 5 ξ2 = (0.1934557, 0.1405538)  X X  O3 5 ξ7 = (0.4695268, 0) We can write θ (д)f as r1i fi ,..., rmi fi , where (rij ) is a O4 5 ξ12 = (0.6260358, 0) i=1 i=1 matrix representation of θ (д). By equation (5) we get m m    −1    −1  X X λ ρ д f1 ,..., λ ρ д fm = λ r1i fi ,..., λ rmi fi , i=1 i=1 * * + * ++ m −1 X , , \ - , -- and therefore ρ(д )fj − rji fi ∈ KerΛ P = ∅ for any 1 ≤ j ≤ i=1  −1 −1  m which implies that f1 ◦ ϑ (д ),..., fm ◦ ϑ (д ) = θ (д)f . □

Example 3.6. The symmetry is given by the representation of the dihedral group D3 in Equation (4). The space Λ of linear forms we consider is spanned by√ the evaluations at the points of the or- Figure 1: Invariant Lagrange interpolation problem and in- √ bits O and O of ξ = (− 5 3 , 1 )t and ξ = (− 3, 1 )t . We define variant interpolant of minimal degree. 1 2 1 3 3 2 3 ϕ : Λ → K2 by ! ! a b ϕ(e ) = ϑ (д) and ϕ(e ) = ϑ (д) . ϑ (д)ξ1 c ϑ (д)ξ2 d The thus defined interpolation problem is clearly ϑ − ϑ equivariant. For each quadruplet (a,b,c,d) ∈ K4 it is desirable to find an inter- t 2 polant (p1,p2) ∈ K[x] that is an ϑ − ϑ equivariant map. This will define the equivariant dynamical system

x˙1(t) = p1(x1(t), x2(t)), x˙2(t) = p2(x1(t), x2(t)) whose integral curves, limit cycles and equilibrium points, will all Figure 2: Graph of a minimal degree interpolant obtained exhibit the D3 symmetry. In Figure 3 we draw the integral cuves of equivariant vector thus constructed. The data of the in- from a monomial basis. The D5 symmetry is not respected. terpolation problem are illustrated by the black arrows : they are t t 3.2 Equivariance the vectors (a,c) and (b,d) , with origin in the points ξ1 and ξ2, Let K[x]m be the of polynomial mappings with m compo- together with their transforms. nents, and let θ : G −→ Aut(Km ) be a linear representation on Km. t A polynomial mapping f = (f1, f2,..., fm ) is called ϑ −θ equivari- ant if f (ϑ (д)x) = θ (д)f (x) for any д ∈ G. The space of equivariant K K θ K G − mappings over , denoted by [x]ϑ , is a [x] module. Equivariant maps define, for instance, dynamical systems that exhibit particularly interesting patterns and are relevant to model physical or biological phenomena [1, 12]. In this context, it is in- teresting to have a tool to offer equivariant maps that interpolate some observed local behaviors. Definition 3.4. Let Λ be a space of linear forms on K[x] and ϕ :   Λ −→ Km a linear map. The pair Λ, ϕ defines a ϑ − θ equivariant interpolation problem if Figure 3: Integral curves for the equivariant vector field in- (1) Λ is closed under the action of G. terpolating the invariant set of 12 vectors drawn in black (2) ϕ(ρ∗(д−1)(λ)) = θ (д)ϕ(λ) for any д ∈ G and λ ∈ Λ.   t The solution of an EIP Λ, ϕ , is a polynomial map f = (f1,..., fm ) 4 SYMMETRY REDUCTION such that λ(f ) = (λ(f ),..., λ(f ))t = ϕ(λ) for any λ ∈ Λ. It is 1 m In this section we show how, when the space Λ of linear forms is natural to seek f as an equivariant map. It is remarkable that any invariant, the Vandermonde matrix can be made block diagonal. type of equivariance will be respected as soon as the interpolation That happens when making use of symmetry adapted bases both space is invariant. for K[x]≤δ and Λ. We start by recalling their general construction, Proposition 3.5. Let (Λ, ϕ) be an equivariant interpolation prob- as it appears in representation theory. The material is drawn from lem. Let P be an invariant interpolation space for Λ and let f = [6, 21]. This block diagonalisation of the Vandermonde indicates ,..., t , ∈ K θ (f1 fm ) be the solution of (Λ ϕ) in P. Then f [x]ϑ . how computation can be organized more efficiently, and robustly. It 4 just draws on the invariance of the space of linear forms. So, when we denote by Pj,α the polynomial map defined by the evaluation points can be chosen, it makes sense to introduce   Pj,α = π (pj ),..., π (pj ) . symmetry among them. j,α1 1 j,α1 cj (10) A symmetry adapted basis P is characterized by the fact that 4.1 Symmetry adapted bases  ∗ [ρ(д)]P = diag R1(д) ⊗ Ic1 ,..., RN (д) ⊗ IcN .Then [ρ (д)]P∗ = A linear representation of the group G on the C− V is a  −t  diag R (д) ⊗ Ic | i = 1..N . group from G to the group GL(V ) of from i i V to itself. V is called the representation space and n is the dimension P ∪N P Proposition 4.1. If = i=1 i be a symmetry adapted basis (or the degree) of the representation ρ. If V has finite dimension n, of V where Pi spans the isotypic component associated to ρi then its and ρ is a linear representation of G on V , upon introducing a basis P∗ ∪N P∗ ∗ dual basis = i=1 i in V is a symmetry adapted basis where P of V the isomorphism ρ(д) can be described by a non-singular ∗ ∗ Pi spans the isotypic component associated to ρi . n × n matrix. This representing matrix is denoted by [ρ(д)]P . The complex-value function χ : G −→ C, with χ (д) → Trace(ρ(д)) is Corollary 4.2. If P is a symmetry adapted basis of K[x]≤δ , so † the character of the representation ρ. is its dual P with respect to the apolar product. The dual or contragredient representation of ρ is the representa- ∗ ∗ A scalar product is G−invariant with respect to a linear represen- tion ρ on the dual vector space V defined by: tation ρ if ⟨v,w⟩ = ρ(д)(v), ρ(д)(w) for any д ∈ G and v,w ∈ V . ∗ −1 ∗ ρ (д)(λ) = λ ◦ ρ(д ) for any λ ∈ V . (6) If we consider unitary representing matrices Ri (д), and an orthonor- ∗ ∗ − j j ∗ 1 t mal basis {p ,...,p } of Vi,1 with respect to a G−invariant inner If P is a basis ofV and P its dual basis then [ρ (д)]P = [ρ(д )]P . 1 cj −1 product, then the same process leads to an orthonormal symmetry It follows that χρ∗ (д) = χρ (д ) = χ ρ (д) A linear representation ρ of a group G on a space V is irreducible adapted basis [6, Theorem 5.4]. if there is no proper subspace W of V with the property that, for Some irreducible representations might not have representing R every д ∈ G, the isomorphism ρ(д) maps every vector of W into W . matrices in . Yet one can determine a real symmetry adapted basis In this case, its representation space V is also called irreducible. The [2] by combining the isotypic components related to conjugate contragredient representation ρ∗ is irreducible when ρ is. A finite irreducible representations. This happens for instance for abelian group has a finite number of inequivalent irreducible representa- groups and we shall avoid them in the examples of this paper for tions. Any representation of a finite group is completely reducible, lack of space. Indeed the completely general statements become meaning that it decomposes into a finite number of irreducible convoluted when working with the distinction. subspaces. 4.2 Block diagonal Vandermonde matrix Let ρj (j = 1,..., N ) be the irreducible nj dimensional represen- tations of G. The complete reduction of the representation ρ and its We consider a linear representation ϑ of a finite group G on Kn. It ∗ representation space are denoted by ρ = c1ρ1 ⊕· · ·⊕cN ρN and V = induces the representations ρ and its dual ρ on the space K[x] and ∗ V1 ⊕ · · · ⊕VN . Each invariant subspace Vj is the direct sum of cj irre- K[x] . K[x]δ is invariant under ρ and thus can be decomposed into LN ducible subspaces and the restriction of ρ to each one is equivalent isotypic components K[x]δ = j=1 Pj , where Pj is associated to to ρj . The (cjnj )−dimensional subspaces Vj of V are the isotypic the irreducible representation ρj of G, with character χj . Each Pj components. With χj the character of ρj we determine the multi- is the image of K[x]δ under the map πj , as defined in (7). ∗ ∗ plicity cj and the projection πj onto the isotypic component Vj For an invariant subspace Λ of K[x] the restriction of ρ to Λ G 1 X nj X −1 is a linear representation of . We shall arrange the isotypic cj = χj (д)χ (д), πj = χj (д )ρ(д). (7) ∗ ⊕ ⊕ ∗ ∗ |G| |G| decomposition Λ = Λ1 ... ΛN such that Λj is the isotypic д ∈G д ∈G ∗ component associated to the irreducible representation ρj , with To go further in the decomposition, consider the representing ∗   character χj . To make a distinction we denote πj,α β as the map = j ≤ , ≤ ∗ matrices Rj (д) rα β (д) for ρj . For 1 α β nj , let defined in (8) associated to ρ . 1≤α, β ≤nj ∗ nj X Proposition 4.3. Let ρ and ρ be linear representations of a finite π = r j (д−1)ρ(д). (8) j,α β |G| βα N д ∈G [ j group G on K[x]≤δ and Λ defined as above. Let P = P be a { j ,..., j } = j=1 Let p1 pcj be a basis of the subspace Vj,1 πj,11(V ).A sym- symmetry adapted basis of K[x]≤δ with metry adapted basis of the isotypic component Vj is then given by •{ j ,..., j } K . j j j j p1 pcj a basis of πj,11( [x]≤δ ) Pj = {p ,...,pc ,..., πj,nj 1(p ),..., πj,nj 1(pc )}. (9) 1 j 1 j •P j = { j ,..., j ,..., j ,..., j }. p1 pcj πj,nj 1(p1) πj,nj 1(pcj ) The union P of the Pj of Vj , is a symmetry adapted basis for V . j j N Indeed, by [21, Proposition 8], the set {π (p ),..., π (p )} [ j j,α1 1 j,α1 cj Let L = L be a symmetry adapted basis of Λ with ⊕ · · · ⊕ is a basis of Vj,α = πj,αα (V ) and Vj = Vj,1 Vj,nj . Fur- j=1 j j j thermore p , π (p ),..., π (p ) is a basis of an irreducible j j k j,21 k j,nj 1 k •{ } ∗   λ ,..., λrj a basis of π (Λ). ( )j 1 j,11 j j j ∗ j ∗ j space with representating matrices rα β (д) . Hereafter •L = {λ ,..., λ ,..., π (λ ),..., π (λ )}. 1≤α, β ≤nj 1 rj j,nj 1 1 j,nj 1 rj 5 √ P − − 3 − − − , Then the Vandermonde matrix WL is given by with λ3 = ϱ1 + ϱ2 ϱ4 ϱ5 λ5 = √2 (ϱ2 ϱ1 + ϱ4 + 2ϱ3 2ϱ6 ϱ5) − − 3 − − − − λ4 = ϱ3 ϱ4 ϱ5 + ϱ6, λ6 = 2 (2ϱ2 2ϱ1 ϱ4 + ϱ3 ϱ5 ϱ6).  j j  ⊗ A symmetry adapted basis of R[x]≤3 is given by diag Inj λs (pt ) 1≤s ≤rj , i = 1 ... N , ≤ ≤ 1 t cj 2 2 3 2 * + [1, x + x , x − 3x1x ] where ⊗ denotes the Kronecker product. 1 2 1 2 , - P := [x 2x − 1 x 3] .  1 2 3 2  Proof. Let α, β,γ, σ ∈ N such that 1 ≤ α, β ≤ nj and 1 ≤    [[x , x 2 − x 2, x 3 + x x 2], [x , −2x x , x 2x + x 3]]  , ≤ j = ∗ j i = i  1 1 2 1 1 2 2 1 2 1 2 2  γ σ ni . Let λα β πj,α1(λβ ) and pγ σ πi,γ 1(pσ ). For any    P  j i P The Vandermonde matrix W is block diagonal :  entry λ (pγ σ ) in WL we have the following: L α β  √  A1 304 − A1 = 0 3 240 3 448 √ √ n X * 9 + 16 3 1216 3 j i j i j i i −1 i WP = . /, − 128 − λ (pγ σ ) = λ (πi,γ 1 (pσ )) = λ r γ (д )ρ (д)(pσ ) L . / 3 3 9 α β α β α β | | 1 . A / G ∈ . 3 / A3 = √ √ . д G . / 2 3 152 3 * + . / * − − 40 − + . / . A3 / . 3 3 9 / . / . / , - n X , - , - = i ri (д−1)ρ∗(д−1)(λj )(pi ) 5 EQUIVARIANT INTERPOLATION |G| 1γ α β σ д ∈G In this section we shall first show how to build interpolation spaces n X = i ri (д)ρ∗(д−1)(λj )(pi ) = π ∗ (λj )(pi ). of minimal degree that are invariant. We shall actually build symme- |G| γ 1 α β σ i,1γ α β σ д ∈G try adapted bases for these, exploiting the block diagonal structure of the Vandermonde matrix. Doing so we prove that the least inter- Using Proposition [21, Proposition 8] (2) if i j, π ∗ (λj ) = 0 , i,1γ α β polation space is invariant. We then present a selection of invariant j i P then λα β (pγ σ ) is zero for i , j, i.e., WL is block diagonal in the or equivariant interpolation problems. As proved in Section 3, the isotypic components. Now if i = j invariance or equivariance is preserved by the interpolant when the interpolation space is invariant. The use of the symmetry adapted λj (pj ) = π ∗ (λj )(pj ) = π ∗ ◦ π ∗ (λj )(pj ). α β γ σ j,1γ α β σ j,1γ j,α1 β σ bases constructed allows this equivariance to be preserved exactly, independently of the numerical accuracy. ∗ j j = ∗ ∗ j j πj,11(λβ )(pσ ) if α γ Since π ◦ π (λ )(pσ ) = , using j,1γ j,α1 β 0 otherwise  5.1 Constructing invariant interpolation spaces the fact that π ∗ (λj ) = λj we get that j,11 β β  The starting point is a representation ϑ of G on Kn that induces  representations ρ and ρ∗ on K[x] and K[x]∗. It is no loss of gener- j j = = j i λβ (pσ ) if i j and α γ λ (pγ σ ) = (11) ality to assume that ϑ is an orthogonal representation. The apolar α β 0 otherwise.  product is thus G-invariant.  ∗ Thus the Vandermonde matrix WP has the announced block Let Λ be an invariant subspace of K[x] . Hereafter L is a symme-  L diagonal structure. □ try adapted basis of Λ and P a symmetry adapted basis of K[x]≤δ consisting of homogeneous polynomials. The elements of P corre- sponding to the same irreducible component are ordered by degree. Remark 1. At the heart of the above proof is the following prop- P ⊗  L According to Proposition 4.3, W = diag Ini Ai . In the fac- V N V V∗ L erty : for a representation = j=1 i of G, and its dual = torization L U := A provided by Gauss elimination, let j , j ,..., j L i i i 1 2 rj N V∗ ∈ V∗ ∈ V ∗ j=1 i , we have λ(v) = 0 as soon as λ i while v j for be the echelon index sequence of Ui ; ri is the multiplicity of ρi in i , j. Λ. An echelon index sequence for Di = Ini ⊗ Ai is given by

Example 4.4. Let G be the dihedral group D3 of order 6. A repre- c[i −1 2 sentation of G on R is given by Equation (4) with m = 3. D3 has Si = {j1 + kni , j2 + kni ,..., jri + kni }. three irreducible representations, two of dimension 1 and one of k=0 dimension 2.  √ t P SN 5 3 1 2 An echelon index sequence of WL is given by S = i=1 Si . Let Consider Ξ the orbit of the point ξ1 = − , in R , with 3 3 Pi be the set of elements of Pi that are indexed by elements of S . e Λ i ξi = ϑi−1ξ1. Let Λ = span( ξ ◦ D ⃗ ) with D ⃗ the directional i ξi ξi From (9) we get that with direction ξ⃗ . Λ is closed under the action ofG. Indeed i Pi { i i i i } ∗ e e −1 = bj ,...,bj ,..., πi,ni 1(bj ),..., πi,ni 1(bj ) . for any p ∈ K[x], ρ (д)( ξ ◦ D ⃗ )(p) = ξ ◦ D ⃗ (p(ϑ (д x)) = Λ 1 ri 1 ri i ξi i ξi − e − 1 ϑ (д 1)ξ ◦D ⃗−1 (p(x)). Since ϑ (д )ξi = ξj for some 1 ≤ j ≤ 6 i ϑ (д )ξi ∗ e e e We prove the assertions made on the outputs of the algorithm. we have ρ (д)( ξ ◦ D ⃗ ) = ξ ◦ D ⃗ . Considering ϱi = ξ ◦ D ⃗ , i ξi j ξj i ξi a symmetry adapted basis of Λ is given by Proposition 5.1. The set of polynomials PΛ built it in Algorithm 1 spans a minimal degree interpolation space for Λ that is invariant [ϱ1 + ϱ2 + ϱ3 + ϱ4 + ϱ5 + ϱ6] P L := [ϱ1 − ϱ2 + ϱ3 − ϱ4 + ϱ5 − ϱ6] , under the action of ρ. Λ is furthermore a symmetry adapted basis  [[λ , λ ], [λ , λ ]]  for this space.  3 4 5 6    6   Algorithm 1 Invariant interpolation space Gauss Elimination by segment as in [4], then HΛ is an orthonormal symmetry adapted basis of Λ↓. In: P and L s.a.b of K[x]≤δ and Λ respectively. With this construction we reproved that Λ is invariant. The Out: - a s.a.b PΛ of an invariant interpolation space of min. degree ↓ above approach to computing a basis of Λ is advantageous in - a symmetry adapted basis HΛ of Λ↓. ↓ P two ways. First Gaussian elimination is performed only on smaller 1: Compute W ; L blocks. But also, when solving invariant and equivariant interpola- 2: for i = 1 to N do   (i) tion problems, the result will respect exactly the intended invariance 3: Ai := Li Ui ; with Ui = u ▷ LU factorization of Ai or equivariance, despite possible numerical inaccuracy. ℓk ℓ,k 4: J := (j1,..., jrj ); ▷ echelon idex sequence of Ui c[i −1 5.2 Computing interpolants ← { } 5: Si j1 + kni , j2 + kni ,..., jri + kni ; We consider an interpolation problem (Λ, ϕ) where Λ is aG-invariant k=0 subspace of K[x]∗ and ϕ : Λ → Km. Take P to be a symmetry Pi ← ∈ Pi ∈ 6: Λ pℓ : pℓ and ℓ Si ; adapted basis of an invariant interpolation space P for Λ as ob- ( ) X tained from Algorithm 1. The interpolant polynomial p that solves i (i) † i 7: H ← u p : pk ∈ P and ℓ ∈ Si ; (Λ, ϕ) in P is given by Λ  ℓk k  d(pk )=d(pℓ )    XN Xni [N  [N  −1 i,α t i,α t i i  p = Ai ϕ(L ) (P ) , (12) 8: PΛ ← P and HΛ ← H ;  Λ Λ i=1 α=1 i=1 i=1 9: P , H i,1 return ( Λ Λ); Pi,α Li,α P where , are as in (10) and Ai = WLi,1 . Note that we made no asumption on ϕ. The invariance of Λ allows to cut the problem into smaller blocks, independently of the structure of ϕ. This illus- Proof. Since the elements of PΛ are indexed by the elements P trate how symmetry can be used to better organize computation : of S then W Λ is invertible and therefore P is an interpolation L Λ if we can choose the points of evaluation, the computational cost space for Λ. The elements of PΛ that correspond to the same P can be alleviated by choosing them with some symmetry. blocks of WL are ordered by degree. Then as a direct consequence When ϕ is invariant or equivariant, Equation (12) can be further of Proposition 2.2, PΛ is a minimal degree interpolation space. reduced. If (Λ, ϕ) is an invariant interpolation problem, it follows We prove now that for any p in PΛ, ρ(д)(p) ∈ PΛ. Considering from Remark 1 that ϕ(Lj ) = 0 for any j > 1. Therefore for solving p = πj,α1(b). By Proposition [21, Proposition 8] (3) we have that any invariant interpolation problem we only need to compute the n Xj P −1 1 t 1 t j first block of WL , i.e., the interpolant is given by A1 ϕ(L ) (P ) . ρ(д)(p) = r (д)π , (b). As π , (b) ∈ PΛ for any 1 ≤ β ≤ nj , βα j β1 j β1 More generally if (Λ, ϕ) is a ϑ − θ equivariant problem, such β=1 that the irreducible representation ρi does not occur in θ, then we conclude that ρ(д)(p) ∈ PΛ. Since PΛ is a basis of PΛ we can ϕ(Li ) = 0. The related block can thus be dismissed. conclude that PΛ is invariant under the action of ρ. □ Example 5.3. Following on Example 3.2. Since we are interested Proposition 5.2. The set HΛ built it in Algorithm 1 is a symmetry in building an interpolation space for an invariant problem, we G G LG adapted basis for Λ↓. only need to compute bases of Λ and K[x]≤5. We have = e , P6 e , P11 e , P16 e PG { , 2 2, 4 ξ1 i=2 ξi i=7 ξi i=12 ξi and = 1 x1 + x2 x1 + Proof. By Proposition 2.3 we get that HΛ is a basis of Λ↓. Let G ( 2 2 + 4, 5 − 3 2 + )4}. = P α j j α T 2x1x2 x2 x1 10x1x2 5x1x2 Since W WLG is a square H = {h ,...,hm ,α } = V HΛ with 1 ≤ α ≤ cj . By the j 1,α j j PG block diagonal structure and Corollary 4.2 we have matrix with full rank, spanK( ) contains a unique invariant in- terpolant for any invariant interpolation problem. It has to be the X   X   least interpolant. hj = u(j)π qj = π u(j)qj = π hj . ℓ,α ℓk j,α1 k j,α1 ℓk k j,α1 ℓ,1 For ϕ given in Example 3.2, one finds the interpolant p by solv- k k G .* /+ ing the 5 × 5 linear system W a = ϕ(L ). The solution a = Therefore H j has the following structure (−0.3333333, 3.295689, −36.59337, 45.36692)t provides the coeffi- Λ , -    cients of PG in p. The graph of p is shown in Figure 1. If p given j j j  j  j H = h ,...,h ,..., π , h ,..., π , h . Λ 1,1 mj,1 j nj 1 1,1 j nj 1 mj,1 above is only an approximation of the least interpolant, due to nu- merical inaccuracy, it is at least exactly invariant. Had we computed Since for any ℓ, h , π (h ),... π (h ) form a basis of an irre- ℓ j,21 ℓ j,nj 1 ℓ the least interpolant with the algorithm of [4], i.e., by elimination ducible representation of G we can conclude that H is a symmetry Λ of the Vandermonde matrix based on the monomial basis, the least adapted basis of Λ. □ interpolant obtained would not be exactly invariant because of the propagation of numerical inacurracies. P α As pointed out in Section 4.1, we can construct a symmetry We define the deviation from invariance (ISD) ofp = deg α ≤5 aα x     P a22 − a32 a14 P | | adapted basis of K[x]δ that is orthonormal with respect to the as σ (a20, a02) + σ a40, 2 , a04 + σ a50, 10 , 5 + β ∈B aβ where σ † apolar product. Then P = P and the basis PΛ built in Algorithm 1 is the standard deviation, and B represents the exponents of the is orthonormal. Moreover if in the third step of Algorithm 1 we use monomials that do not belongs to any of the elements in PG . In 7 √ λ = e + e − e − e λ = 3 (−e + e + 2e + e − e − 2e ), Table 1 we show the ISD for the interpolant p computed with dif- with 1 ξ1 ξ2 ξ4 ξ5 5 √2 ξ1 ξ2 ξ3 ξ4 ξ5 ξ6 λ = e − e − e + e , λ = 3 (−2e + 2e + e − e − e − e ). ferent precisions. The obtained polynomials are somehow far from 2 ξ3 ξ4 ξ5 ξ6 6 √2 ξ1 ξ2 ξ3 ξ4 ξ5 ξ6 − λ = e + e − e − e λ = 3 (−e + e + 2e + e − e − 2e ), being G invariant. 3 ξ7 ξ8 ξ10 ξ11 7 √2 ξ7 ξ8 ξ9 ξ10 ξ11 ξ12 λ = e − e − e + e , λ = 3 (−2e + 2e + e − e − e − e ). 4 ξ9 ξ10 ξ11 ξ12 8 2 ξ7 ξ8 ξ9 ξ10 ξ11 ξ12 # Digits 10 15 20 30 The columns correspond to −9 ISD 72.9614 40.0289 6.0967 < 10 P3,1 := x, x 2 − y2, x 3 + xy2, x 4 − y4 , 3 Table 1: ISD values for different digits of precision P := f g . P3,2 − 2 2 − 2 2  := y, 2xy, y(x + y ), 2xy(x + y )   f √ √ g   72 3 −288 608 3 −2432   √ √  2 9 3 90 76 3 760 In the same spirit, let us mention that the condition number of A = − √ √ 3 27 45 3 −90 140 3 280 M K 2 * √ √ + WΛ , where M is the monomial basis of [x]≤5, is more than 10 . 9 3 18 28 3 504 / PG . / times the condition number of WLG . This is an indicator that two We thus determine that the equivariant interpolant for the interpo- additional digits of precision are lost in the computation. lation problem described, in Example 3.6 is given by : -

α 3 β 2 2 9γ 2 2 27 δ 4 4 Example 5.4. Following up on Example 4.4. Let θ be the permuta- p1 = x + (x − y ) + x (x + y ) + (x − y ) 3 tion representation of D3 in R . θ decomposes into two irreducible 320 640 8960 17920 representations, the trivial representation and the irreducible rep- α 3 β 9γ 2 2 27 δ 2 2 p2 = y − xy + y(x + y ) − xy(x + y ) resentation ϑ of dimension 2. Let ϕ : Λ → R3 a ϑ − θ equivariant 320 320 8960 8960 t where map determined by ϕ(ϱ1) = (1, −1, 5) . For solving (Λ, ϕ) we need √ √ only consider the first and third block of the Vandermonde matrix α =√3(25 a − 114 b) + 494 d − 185 c, β =√ 3(114 d − 25 c) + 38 b − 5 a, computed in Example 4.4. The ρ∗ − θ equivariant map that solve γ = 3(42 b − 25 a) + 185 c − 182 d, δ = 3(25 c − 42 d) + 5 a − 14 b. (Λ, ϕ) is P = (p1,p2,p3) with: √ √ 705 2 135 2 31 93 15 REFERENCES p1 := x1 + x2 + 3x1 + x2 − 3x1x2 4256 4256 56 56 112 [1] P. Chossat and R. Lauterbach. 2000. Methods in equivariant bifurcations and √ √ dynamical systems. World Scientific Publishing. 705 2 135 2 31 93 15 p2 := x1 + x2 + 3x1 − x2 + 3x1x2 [2] M. Collowald and E. Hubert. 2015. A moment matrix approach to computing 4256 4256 56 56 112 symmetric cubatures. (2015). https://hal.inria.fr/hal-01188290. √ 75 2 495 2 31 [3] C. De Boor and A. Ron. 1990. On multivariate polynomial interpolation. Con- p3 := − x1 + x2 − 3x1. structive Approximation 6, 3 (1990). 2128 2128 28 [4] C. De Boor and A. Ron. 1992. Computational aspects of polynomial interpolation In Figure 4 we show the image of R2 by P and the tangency condi- in several variables. Math. Comp. 58, 198 (1992). tions imposed by ϕ. [5] C. De Boor and A. Ron. 1992. The least solution for the polynomial interpolation problem. Mathematische Zeitschrift 210, 1 (1992). [6] A. Fässler and E. Stiefel. 1992. Group theoretical methods and their applications. Springer. [7] J.-C. Faugère and S. Rahmany. 2009. Solving systems of polynomial equations with symmetries using SAGBI-Gröbner bases. In Proc. ISSAC 2009. ACM, 151–158. [8] J.-C. Faugere and J. Svartz. 2013. Grobner bases of ideals invariant under a commutative group: the non-modular case. In Proc. ISSAC 2013. ACM, 347–354. [9] M. Gasca and T. Sauer. 2000. Polynomial interpolation in several variables. Advances in Computational Mathematics 12, 4 (2000), 377. [10] K. Gatermann. 2000. Computer algebra methods for equivariant dynamical systems. Lecture Notes in Mathematics, Vol. 1728. Springer-Verlag, Berlin. [11] K. Gatermann and P. A. Parrilo. 2004. Symmetry groups, semidefinite programs, and sums of squares. J. Pure Appl. Algebra 192, 1-3 (2004), 95–128. [12] M. Golubitsky, I. Stewart, and D. G. Schaeffer. 1988. Singularities and groups in bifurcation theory. Vol. II. Applied Mathematical Sciences, Vol. 69. Springer. [13] E. Hubert. 2019. Invariant Algebraic Sets and Symmetrization of Polynomial Systems. Journal of Symbolic Computation (2019). DOI:10.1016/j.jsc.2018.09.002. [14] E. Hubert and G. Labahn. 2012. Rational invariants of scalings from Hermite normal forms. In Proc. ISSAC 2012. ACM, 219–226. [15] E. Hubert and G. Labahn. 2016. Computation of the Invariants of Finite Abelian Figure 4: Parameterized surface with tangency constraints. Groups. Mathematics of Computations 85, 302 (2016), 3029–3050. [16] T. Krick, A. Szanto, and M. Valdettaro. 2017. Symmetric interpolation, Exchange Lemma and Sylvester sums. Comm. Algebra 45, 8 (2017), 3231–3250. [17] RA Lorentz. 2000. Multivariate Hermite interpolation by algebraic polynomials: Example 5.5. Following up on Example 3.6. Since the representa- A survey. Journal of computational and applied mathematics 122, 1-2 (2000). tion ϑ of D in R2 is irreducible, for computing any ϑ −ϑ equivariant [18] C. Riener and M. Safey El Din. 2018. Real root finding for equivariant semi- 3 algebraic systems. In Proc. ISSAC 2018. ACM, 335–342. we only need to compute the third isotopic block in the Vander- [19] C. Riener, T. Theobald, L. J. Andrén, and J. B. Lasserre. 2013. Exploiting symmetries P3 P in SDP-relaxations for polynomial optimization. Math. Oper. Res. 38, 1 (2013). monde matrix WL3 , where is a basis for the interpolation space ! [20] T. Sauer. 1998. Polynomial interpolation of minimal degree and Gröbner bases. A3 London Mathematical Society Lecture Note Series (1998). PΛ built by Algorithm 1. This block is W = . The rows Linear representations of finite groups A3 [21] J. P. Serre. 1977. . Springer. [22] J. Verschelde and K. Gatermann. 1995. Symmetric Newton polytopes for solving correspond to sparse polynomial systems. Adv. in Appl. Math. 16, 1 (1995), 95–127. 3 3,1 3,2 3,1 3,2 L := L , L , L := [λ1, λ2, λ3, λ4] and L := [λ5, λ6, λ7, λ8] f g 8