Ideal Interpolation, H-Bases and Symmetry Erick Rodriguez Bazan, Evelyne Hubert

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Erick Rodriguez Bazan, Evelyne Hubert. Ideal Interpolation, H-Bases and Symmetry. ISSAC 2020 - International Symposium on Symbolic and Algebraic Computation, Jul 2020, Kalamata, Greece. ￿10.1145/3373207.3404057￿. ￿hal-02482098v2￿

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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 for each instantiated interpolation problem, that is both invariant Multivariate Lagrange and Hermite interpolation are examples of and of minimal degree. An interpolation space for Λ identifies with ideal interpolation. More generally an ideal interpolation problem the quotient space K[x]/I. Hence a number of operations related is defined by a set of linear forms, on the polynomial ring, whose to I can already be performed with a basis of an interpolation kernels intersect into an ideal. space for Λ: decide of membership to I, determine normal forms For an ideal interpolation problem with symmetry, we address of polynomials modulo I and compute matrices of multiplication the simultaneous computation of a symmetry adapted basis of the maps in K[x]/I. Yet it has also proved relevant to compute Gröbner least interpolation space and the symmetry adapted H-basis of bases or H-bases of I. the ideal. Beside its manifest presence in the output, symmetry is Initiated in [26], for a set Λ of point evaluations, computing a exploited computationally at all stages of the algorithm. Gröbner basis of I found applications in the design of experiments [29, 30]. As pointed out in [25], one can furthermore interpret the CCS CONCEPTS FGLM algorithm [10] as an instance of this problem. The linear forms are the coefficients, in the normal forms, of the reduced • Computing methodologies → Symbolic and algebraic algo- monomials. The alternative approach in [11] can be understood rithms; similarly. KEYWORDS The resulting algorithm then pertains to the Berlekamp-Massey- Interpolation; Symmetry; Representation Theory; Group Action; Sakata algorithm and is related the multivariate version of Prony’s H-basis; Macaulay matrix; Vandermonde matrix problem to compute Gröbner bases, border bases, or H-bases [1, 28, 35, 36] 1 INTRODUCTION All ,the above mentioned algorithms and complexity analyses Preserving and exploiting symmetry in algebraic computations is a heavily depend on a term order and basis of monomials. These challenge that has been addressed within a few topics and, mostly, are notoriously not suited for preserving symmetry. Our ambition for specific groups of symmetry; For instance interpolation and in this paper is to showcase how symmetry can be embedded in symmetric group [23], cubature [4, 14], global optimisation [17, 32], the representation of both the interpolation space and the repre- equivariant dynamical systems [15, 20] and solving systems of sentation of the ideal. This is a marker for the more canonical polynomial equations [12, 13, 16, 19, 21, 31, 38]. In [33] we addressed representations. multivariate interpolation and in this article we go further with ideal The least interpolation space, defined in6 [ ], and revisited in [33] interpolation. We provide an algorithm to compute simultaneously is a canonically defined interpolation space. It serves here as the a symmetry adapted basis of the least interpolation space and a canonical representation of the quotient of the polynomial algebra symmetry adapted H-basis of the associated ideal. In addition to by the ideal. It has great properties, even beyond symmetry, that being manifest in the output, symmetry is exploited all along the cannot be achieved by a space spanned by monomials. In [33] algorithm to reduce the size of the matrices involved, and avoid we freed the computation of the least interpolation space from sizable redundancies. Based on QR-decomposition (as opposed to its reliance on the monomial basis by introducing dual bases. We LU-decomposition previously) the algorithm also lends itself to pursue this approach here for the representation of the ideal by H- numerical computations. bases [24, 27]. Where Gröbner bases single out leading terms with a Multivariate Lagrange, and Hermite, interpolation are examples term order, H-bases work with leading forms and the orthogonality of the encompassing notion of ideal interpolation, introduced in [2]. with respect to the apolar product. The least interpolation space They are defined by linear forms consisting of evaluation atsome then reveals itself as the orthogonal complement of the ideal of nodes, and possibly composed with differential operators, without leading forms. gaps. More generally a space of linear forms Λ on the polynomial As a result, computing a H-basis of the interpolation ideal is achieved with linear algebra in subspaces of homogeneous polyno- ring K[x] = K[x1,..., xn] is an ideal interpolation scheme if mials of growing degrees. Yet we shall first redefine the concepts at Ù I = ker λ = {p ∈ K[x] : λ(p) = 0, for all λ in Λ} (1) play in an intrinsic manner, contrary to the computation centered λ ∈Λ approach in [27, 34]. The precise algorithm we shall offer to com- is an ideal in K[x]. In the case of Lagrange interpolation, I is the pute H-bases somehow fits in the loose sketch proposed5 in[ ]. Yet ideal of the nodes and is thus a radical ideal. we are now in a position to incorporate symmetry in a natural way, If Λ is invariant under the action of a group G, then so is I. In refining the algorithm to exploit it; A totally original contribution. [33] we addressed the computation of an interpolation space for Λ Symmetry is preserved and exploited thanks to the block diago- i.e., a subspace of the polynomial ring that has a unique interpolant nal structure of the matrices at play in the algorithms. This block diagonalisation, with predicted repetitions in the blocks, happens is invertible. This latter is to be interpreted as the matrix in the when the underlying maps are discovered to be equivariant and bases P and the dual of L of the restriction of the Vandermonde expressed in the related symmetry adapted bases. The case of the operator w : K[x] → Λ∗ such that w(p)(λ) = λ(p). This is the adjoint Vandermonde matrix was settled in [33]. In this paper, we also need of embedding Λ ,→ K[x]∗ and hence is surjective. the matrix of the prolongation map, knowned in the monomial All along this paper we shall assume that basis as the Macaulay matrix. Figuring out the equivariance of this I = kerw = ∩λ ∈Λ ker λ map is one of the original key results of this paper. Λ = ⟨e ,..., e ⟩ I The paper is organized as follows. In Section 2 we define ideal is an ideal. When for instance ξ1 ξr K then is the ideal of the points {ξ1,..., ξr } ⊂ K[x]. One sees in general that interpolation and explain the identification of an interpolation ∗ space with the quotient algebra. In Section 3 we review H-bases and dim K[x]/I = dim Λ = dim Λ =: r. discuss how they can be computed in the ideal interpolation setting. With Q = {q1,..., qr } ⊂ K[x], we can identify K[x]/I with In Section 4 we provide an algorithm to compute simultaneously a ⟨Q⟩K if ⟨Q⟩K ⊕ I = K[x]. With a slight shortcut, we say that Q is basis of the least interpolation space and an orthogonal H-basis of a basis for K[x]/I. the ideal. In Section 5 we show how the Macaulay matrix can be Proposition 2.1. Q = {q1,..., qr } ⊂ K[x] spans an interpola- block diagonalized in the presence of symmetry. This is then applied tion space for Λ iff it is a basis for the quotient K[x]/I. in Section 6 to obtain an algorithm to compute simultaneously a symmetry adapted basis of the least interpolation space and a Proof. If Q = {q1,..., qr } is a basis of K[x]/I then for any symmetry adapted H-basis of the ideal. All along the paper, the p ∈ K[x] there is a q ∈ ⟨q1,..., qr ⟩K such that p ≡ q mod I. Hence definitions and notations comply with those in [33]. λ(p) = λ(q) for any λ ∈ Λ and thus ⟨Q⟩K is an interpolation space for Λ. Conversely if ⟨q1,..., qr ⟩K is an interpolation space for Λ 2 IDEAL INTERPOLATION then {q1,..., qr } are linearly independent modulo I and therefore a basis for K[x]/I. Indeed if q = a1q1 + ... + ar qr ∈ I then any In this section, we consider the ideal interpolation problem and interpolation problem has multiple solutions in ⟨Q⟩K, i.e, if p is explain the identification of an interpolation space with the quotient the solution of (Λ, ϕ) so is p + q, contradicting the interpolation algebra. We recall that the least interpolation space is the orthogonal uniqueness on ⟨Q⟩K. □ complement of the ideal of the leading forms, I0. K denotes either C or R. K[x] = K[x ,..., x ] denotes the ring 1 n For p ∈ K[x] we can find its natural projection on K[x]/I by of polynomials in the variables x1,..., xn with coefficients in K; taking the unique q ∈ ⟨Q⟩K that satisfies λ(q) = λ(p) for all λ ∈ Λ. K[x]≤d and K[x]d the K−vector spaces of polynomials of degree From a computational point of view, q is obtained by solving the at most d and the space of homogeneous polynomials of degree d Vandermonde system, i.e., respectively. The dual of K[x], the set of K−linear forms on K[x], λ (p) [ ]∗ [ ] 1 is denoted by K x . A typical example of a linear form on K x is  −1 © . ª n q = (q ,..., q ) WQ ­ . ® with L = {λ ,..., λ } a basis of Λ. the evaluation eξ at a point ξ of K : eξ (p) = p(ξ ). 1 r L ­ . ® 1 r ∗ ­ ® K[x] can be identified with the ring of formal power series λr (p) β α « ¬ K[[∂]] = K[[∂1, . . . , ∂r ]], with the understanding that ∂ (x ) = α! Similarly, the matrix of the multiplication map, in the basis Q, is = K[ ] or 0 according to whether α β or not. Concomitantly x is m K[ ]/I → K[ ]/I Í α p : x x , equipped with the apolar product that is defined, for p = α pα x [q] 7→ [pq] = Í α ⟨ , ⟩ = (∂) = Í ∈ K and q α qα x , by p q : p q α α!pα qα . − †  Q  1 Q If P is a (homogeneous) basis of K[x] we denote P its dual is obtained as [mp ]Q = W W where L ◦ mp = {λ1 ◦ ∗ L L◦mp with respect to this scalar product. For λ ∈ K[x] we can write m ,..., λ ◦ m }. Í † p r p λ = p ∈P λ(p)p (∂). When working with Gröbner bases, one fixes a term order and An interpolation problem is a pair (Λ, ϕ) where Λ is a finite dimen- focuses on leading terms of polynomials and the initial ideal of ∗ sional linear subspace of K[x] and ϕ : Λ −→ K is a K-linear map. I. The basis of choice for K[x]/I consists of the monomials that An interpolant, i.e., a solution to the interpolation problem, is a do not belong to the initial ideal. An H-basis of I is somehow ( ) ( ) ∈ polynomial p such that λ p = ϕ λ for any λ Λ. An interpolation the complement of the least interpolation space Λ↓ and hence can space for Λ is a polynomial subspace P of K[x] such that there is a be made to reflect the possible invariance of Λ and I. Instead of unique interpolant for any map ϕ. leading terms, the focus is then on the leading homogeneous forms. The least interpolation space Λ↓ was introduced in [7], and revis- Hereafter we denote by p0 the leading homogeneous form of p, ited in [33]. The least term λ↓ ∈ K[x] of a power series λ ∈ K[[∂]] is i.e., the unique homogeneous polynomial such that deg p − p0
< the unique homogeneous polynomial for which λ − λ↓(∂) vanishes deg (p). Given a set of polynomials P we denote P0 = p0 | p ∈ P . to highest possible order at the origin. Given a linear space of linear forms Λ, we define Λ↓ as the linear span of all λ↓ with λ ∈ Λ. Proposition 2.2. Let Q be an interpolation space of minimal degree for . Then ⊕ I0 = K[ ]. If L = {λ1, λ2,..., λr } is a basis of Λ and P = {p1,p2,...,pr } ⊂ Λ Q x [ ] P K x , then is a basis for an interpolation space of Λ if and only if Proof. We proceed by induction on the degree, i.e, we assume the Vandermonde matrix that any polynomial p in K[x]≤d can be written as p = q + l where 0 P 
 q ∈ Q and l ∈ I . Note that the hypothesis holds trivially when d WL := λi pj 1≤i ≤r (2) 1≤j ≤r is equal to zero. 2 ∈ K[ ] K[ ] = ⟨Q⟩ ⊕I ∈ 0 Now let p x ≤d+1. Since x K there exists q Q We shall use the notation Pd for the set of the degree d elements ∈ I 0 0 0 and l such that p = q + l. Since Q is of minimal degree, q and of P . In other words P = P ∩ K[x]d . 0 0 d l are in K[x]≤d+1. Writing l = l + l1 he have p = q + l + l1 with ∈ K[ ] ∈ ∈ I0 Definition 3.4. We say that an H-basis H is minimal if, for any l1 x ≤d then by induction l1 = q1 +l2 with q1 Q and l2 0 0 0 d ∈ N, H is linearly independent and and therefore p = q + q1 + l + l2 ∈ Q ⊕ I . □ d   I0 ⊕ H0 I0 Vd d−1 d K = d . (3) As a consequence we retrieve the result of [7, Theorem 4.8]. D E Furthermore H is said to be orthogonal if H 0 is the orthogonal d K Corollary 2.3. Considering orthogonality with respect to the   ⊥ complement of V I0 in I0. 0 d d−1 d apolar product it holds that Λ↓ ⊕ I = K[x]. Note that if hi and hj are two elements with deghi > deghj of 0 Proof. Follows from the fact that λ(p) = 0 ⇒ ⟨λ↓,p ⟩ = 0. □ an orthogonal H-basis we have D E 0 0 ∈ K[ ] hi ,phj = 0 for all p x deg hi −deg hj . 3 H-BASES Definition 3.5. Let H = {h1,..., hm } be an orthogonal H-basis H-bases were introduced by [24]. The use of H-basis in interpolation of an ideal I. The reduced H-basis of H is defined by has been further studied in [27, 34]. In this section we review the n o H 0 − 0 0 − 0 definitions and present the sketch of an algorithm to compute the e = h1 he1,..., hm hfm (4) H-basis of I = Ñ ker λ. λ ∈Λ where, for p ∈ K[x], p˜ is the projection of p on the orthogonal 0 Definition 3.1. A finite set H := {h1,..., hm } ⊂ K[x] is an complement of I parallel to I. H-basis of the ideal I := ⟨h ,..., h ⟩ if, for all p ∈ I there are 1 m [27, Lemma 6.2] show how p˜ can be computed given H. д1,... дm such that, m Schematic computation of H-bases. In the next section we elabo- Õ p = hiдi and deg(hi ) + deg(дi ) ≤ deg(p), i = 1,...,m. rate on an algorithm to compute concomitantly the least interpola- i=1 tion space and an H-basis for the ideal associated to a set of linear forms Λ. As a way of introduction we reproduce the sketch of an Theorem 3.2. [27] Let H := {h1,..., hm } and I := ⟨H⟩. Then the following conditions are equivalent: algorithm as proposed by [5] to compute an H-basis until degree D. It is based on the asumption that we have access to a basis of (1) H is an H-basis of I. I := I ∩ K[x] for any d. I0  0 | ∈ I 0 0 d ≤d (2) := h h = h1,..., hm . Hilbert Basis Theorem says that I0 has a finite basis, hence Algorithm 1 [5] H-basis construction any ideal in K[x] has a finite H-basis. We shall now introduce Input: - a degree D. the concepts of minimal, orthogonal and reduced H-basis. The - basis for Id for 1 ≤ d ≤ D. notion of orthogonality is considered w.r.t the apolar product. Our Output : - an H-basis until degree D definitions somewhat differ from [27] as we dissociate them from 1: H ← {} ; the computational aspect. We need to introduce first the following 2: for d = 0 to D do 0 vector space of homogeneous polynomials. 3: Cd ← a basis of Vd (H ); B ← (H) I0 4: d a basis for the complement of Vd in d ; Definition 3.3. Given a set H = {h1,..., hm } of homogeneous 5: B ← projection of B in I polynomials in K[x] and a degree d, we define the subspace V (H) bd d d d H ← H Ð B as 6: bd ; ( s ) Õ 7: return H; (H) ∈ [ ] ⊂ [ ] Vd = дi hi дi K x d−deg(hi ) K x d . i=1 The correctness of Algorithm 1 is shown by induction. Assume Vd (H) is the image of the linear map ψd : that Hd−1 consists of the polynomials in an H-basis of I up to [ ] × × [ ] → [ ] − ∈ I ( ) ψd,h : K x d−d1 ... K x d−dm K x d degreed 1. Considerp with deg p = d. By Step 4 in Algorithm m Õ . 1 we have (д1,..., дm ) → дi hi 0 Õ 0 Õ p = hi дi + ai bi (5) i=1 hi ∈H bi ∈Bd We denote by MM P (H) the matrix of ψ in the bases M d , d d d with дi ∈ K[x]d−deg(h ) and ai ∈ K. From (5) we have that p ∈ I and P of K[x] × ... × K[x] and K[x] respectively. It is i d d−d1 d−dm d and Í h д + Í a bˆ ∈ I have the same leading form. referred to as the Macaulay matrix for H. We can write V (H) as hi ∈H i i bi ∈Bd+1 i i d Thus Õ Õ ˆ p − hi дi − ai bi ∈ Id−1  |Pd |   Õ    hi ∈Hd−1 bi ∈Bd V (H) = ai pi (a1,..., a ) ∈ R MM ,P (H) , d |Pd | d d therefore using the induction hypothesis we get that  i=0    Õ Õ Õ   p = hiдi + aibˆi + hiqi where R MM , P (H) denotes the column space of MM , P (H). d d d d hi ∈Hd−1 bi ∈Bd+1 hi ∈Hd−1 3 ∈ [ ] H with qi K x ≤d−1−deg(hi ) and therefore is an H-basis. Definition 4.1. Given a space of linear forms Λ, we denote by Algorithm 1 can be applied in the ideal interpolation scheme. In Λ≥d the subspace of Λ given by this setting a basis of I can be computed for any d using Linear d Λ = λ ∈ Λ | λ ∈ K[x] ∪ {0}. Algebra techniques due to the following relation. ≥d ↓ ⩾d Hereafter we organize the elements of the bases of K[x], Λ, or |P | †  ≤d t  their subspaces, as row vectors. In particular P and P are dual  Õ    P≤d   Id = ai pi | a1,..., a |P | ∈ ker WL and pi ∈ P≤d , homogeneous bases for K[x] according to the apolar product. Their  ≤d   i=1  P P† K[ ]   degree part d and d are dual bases of x d . for any basis P≤d of K[x]≤d . A basis L≥d of Λ≥d can be computed inductively thanks to the In the next section we will give an efficient and detailed version following observation. of Algorithm 1 in the ideal interpolation case. We will integrate the Proposition 4.2. Assume L is a basis of Λ . Consider the computations of an H-basis for I = ∩ ker λ and a basis for Λ . ≥d ≥d λ ∈Λ ↓ QR-decomposition When the ideal is given by a set of generators it is also possible   to compute an H-basis with linear algebra if you know a bound on Pd Rd W = [Q1 | Q2 ] · the degree of the syzygies of the generators. A numerical approach, L≥d 0 using singular value decomposition, was introduced in [22]. Alter- and the related change of basis [Ld | L≥d+1] = L≥d · [Q1 | Q2 ]. Then natively an extension of Buchberger’s algorithm is presented in •L ≥d+1 is a basis of Λ≥d+1; [27]. It relies, at each step, on the computation of a basis for the P • R = W d has full row rank; module of syzygies of a set of homogeneous polynomials. d Ld • L = P† · T ∩K[ ] The components of d ↓ d Rd form a basis of Λ↓ x d . 4 SIMULTANEOUS COMPUTATION OF THE H-BASIS AND LEAST INTERPOLATION Ðd We shall furthermore denote by L≤ = Li the thus con- SPACE d i=0 structed basis of a complement of Λ≥d+1 in Λ. In this section we present an algorithm to compute both a (or- ′ thogonal) basis of Λ↓ and an orthogonal H-basis H of the ideal Proof. It all follows from the fact that a change of basis L = P T P I = ∩λ ∈Λ ker λ. We proceed degree by degree. At each iteration LQ of Λ implies that WL′ = Q WL . In the present case Q = T −1 of the algorithm we compute a basis of Λ↓ ∩ K[x]d and the set [Q1 | Q2 ] is orthogonal and hence Q = Q . H 0 = H 0 ∩ K[ ] ∈ d x d . Recall from Corollary 2.3, Theorem 3.2, and The last point simply follows from the fact that, for λ Λ, Í † P Definition 3.4 that λ = p ∈P λ(p)p (∂). Hence if T = WL then the j-th component ⊥   ⊥ L Í †( ) K[ ] ⊕ I0 I0 ⟨H 0⟩ I0 I0 ⊕ H 0 of is i tjip ∂ . □ x = Λ↓ , = , and d = Vd d−1 d K .

I is the kernel of the Vandermonde operator while Λ↓ can be inferred from a rank revealing form of the Vandermonde matrix. This construction gives us a basis of Λ↓ ∩ K[x]d in addition to With orthogonality prevailing in the objects we compute it is natural a basis of Λ≥d+1 to pursue the computation at the next degree. H 0 that the QR-decomposition plays a central role in our algorithm. Before going there, we need to compute a basis d for the com- × (H 0 ) I0 For a m n matrix M, the QR-decomposition is M = QR where Q plement of Vd

Proof. The columns of MM , P† are the vectors of coefficients, d d that V (I0 ) = I0 hence ⟨H 0⟩ is an empty set. The latter implies P† (H) d d−1 d d in the basis d , of polynomials that span Vd . The member- that when the algorithm stops we have computed the full H-basis (H) ship of q in the left kernel of MM P† translates as the apolar H 0 I0 d , d for . product of q with these vectors to be zero. And conversely. □ We then obtain an H-basis of I by finding the projections, onto 0 Λ↓ and parallel to I, of the elements of H . These are the polyno- 0 Proposition 4.5. Consider the QR-decomposition mials of Λ↓ interpolating the elements of H according to Λ.  T     Pd  R Termination. Considering r := dim(Λ) we have that L≥ is an W M † (H) = [ | ] · r L M ,P Q1 Q2 d d d 0 empty set, this implies that in the worst case our algorithm stops The components of the row vector Pd · Q2 span the orthogonal com- after r iterations. plement of V (H) in I0. d d Complexity. The most expensive computational step in Algo-  T   t  Pd  Pd rithms 2 is the computation of the kernel of WL M † (H) , ∩ d Md ,P Proof. The columns in Q2 span ker WL ker MM , P† . d d d d with number of columns and rows given by The result thus follows from Lemmas 4.3 and 4.4. □

n−1 row(d) = d+n−1
= d + O dn−1
We are now able to show the correctness and termination of n−1 (n−1)! (8) Algorithm 2. |H | |H | n−1 ( ) Í d−di +n−1
|L | d O n−1
col d = i=1 n−1 + d = (n−1)! + d Correctness. In the spirit of Algorithm 1, Algorithm 2 proceeds degree by degree. At the iteration for degree d we first compute a where d1,...,d |H | are the degrees of the elements of the com- basis for Λ≥d+1 by splitting L≥d into L≥d+1 and Ld . As explained puted H-basis until degree d. Then the computational complexity of in Proposition 4.2, this is obtained through the QR-decomposition Algorithm 2 relies on the method used for the kernel computation Pd of W . From this decomposition we also obtain a basis for Λ↓ ∩ of VM(d), which in our case is the QR-decomposition. L≥d Pd 0 We are giving a frame for the simultaneous computation of an K[x]d as well as W . We then go after H , which spans the Ld d H-basis and the Least interpolation space, but there is still room (H 0 ) I0 H 0 orthogonal complement of Vd ≤d−1 in d . The elements of d for improving the performance of Algorithm 2. The structure of  t Pd the Macaulay matrix might be taken into account to alleviate the are computed via intersection of ker WL and ker MM P† as d d , d linear algebra operations as for instance in [1]. We can also consider showed in Proposition 4.5. Algorithm 2 stops when we reach a different variants of Algorithm 2. In Proposition 4.6 we show that 0 degree δ such that L≥δ is empty. Notice that for d ≥ δ the matrix orthogonal bases for K[x]d ∩ Λ↓ and I can be simultaneously P d W d is an empty matrix and therefore its kernel is the full space computed by applying QR-decomposition in the Vandermonde Ld P K[ ] d > δ matrix (W d )T . Therefore we can split Step 9 in two steps. First x d . Then as a consequence of Lemma 4.3, for all we have L≥d 5 P we do a QR-decomposition (W d )T to obtain orthogonal bases of the Macaulay matrix when the space spanned by H is invariant L≥d K[ ] ∩ I0 I0 under the induced action of a group G on K[x]. The key relies on x d Λ↓ and d . Once that we have in hand a basis of d we exhibiting the equivariance of the prolongation map Ψ , defined obtain the elements of Hd as its complement in the column space d h (H) in Section 3. of MM , P† . d d With notations compliant with [33], for any representation θ of   T a group G on a K-vector space V , a symmetry adapted basis P of V Rd  Pd  Proposition 4.6. Let [Q1 | Q2] · = W be a QR- is characterized by the fact that the matrix of the representation θ 0 L≥d P  P T  P T in is decomposition of W d . Letr be the rank of W d . Let {q ... q } L≥ L≥ 1 r
d d [θ (д)]P = diag R1(д) ⊗ Ic1 ,..., RN (д) ⊗ IcN . and {q ... q } be the columns of Q and Q respectively. Then r +1 m 1 2  j  where Rj = r is the matrix representation of the irre- the following holds: kl 1≤k,l ≤n n o j P P† · ,..., P† · K[ ] Ñ ducible representation ρj of G and cj is the multiplicity of ρj in (1) Λ,d = d q1 d qr is a basis of x d Λ↓. P ∪N Pj Pj 0 θ. Hence = j=1 where spans the isotypic component Vj (2) N = {Pd · qr +1,..., Pd · qm } is a basis of I . d nj Í j ( −1) ( ) ⊥ associated to ρj . Introducing the map πj,kl = |G | д∈G rkl д θ д (3) Ifp ∈ P andq ∈ N then ⟨p, q⟩ = 0, i.e., K[x] = Λ ∩ K[x]
⊕ Λ,d ↓ d we can say that Pj is determined by p j ,..., p j to mean that I0. 1 cj d p j ,..., p j is a basis of π (V ) and In the case where P is orthonormal with respect to the apolar product, 1 cj j,11 † Pj { j j ( j ) ( j )} i.e. P = P , then PΛ,d and N are also orthonormal bases. = p1 ,..., pcj ,..., πj,nj 1 p1 ,..., πj,nj 1 pcj . (11) Ð When dealing with K = R, the statements we write are for the Proof. Let D such that L≥D = {} and let L≤D = d ≤D Ld P≤D case where all the irreducible representations of G are absolutely be a basis of Λ. Then the matrix WL is block upper triangular ≤D irreducible, and thus the matrices Rj (д) all have real entries. This |P≤ | with non singular diagonal blocks. Consider {a1,... aℓ } ∈ K D is the case of all reflection groups. Yet these statements canbe P the rows of W ≤D . By Proposition [33, Proposition 2.3] we have modified to also work with irreducible representations of complex L≤D    † t   † t  type, which occur, for instance, for the cyclic group Cm with m > 2. that PΛ P · a ,..., P · a is a basis of Λ↓, we can ≤D 1 ↓ ≤D ℓ ↓ Consider now a set H = {h1,..., hl } of homogeneous polyno- D mials of K[x]. We denote d ,...,d their respective degrees and Ø n o 1 ℓ rewrite P as P† · bt ,..., P† · bt where b ,...,b is a ℓ Λ d 1 d ℓd 1 ℓd h = [h1,..., hℓ] the row vector of K[x] . Associated to h, and a d=1 degree d, is the map introduced in Section 3  Pd  basis of the row space of W . Since PΛ is a graded basis then Ld ψd,h : K[x]d−d × ... × K[x]d−d → K[x]d n o 1 ℓ P† · t ,..., P† · t K[ ] ∩ (12) d b1 d bℓ is a basis x d Λ↓. t d f = [f1,..., fℓ ] → h · f. Part (2) in the proposition is a direct consequence of Lemma 4.3 and the fact that the columns of Q2 form a basis of the kernel of We assume that H forms a basis of an invariant subspace of Pd W . Let now q ∈ P , and p ∈ N. Then, K[x] and we call θ the restriction of the representation ρ to this L≥d Λ d * + subspace, while Θ is the matrix representation in the basis H: Θ(д) = Õ Õ Õ −1 ⟨p, q⟩ = ai pi , bi qi = ai bi = 0. [θ (д)]H. Then [ρ(д)(h1),..., ρ(д)(hℓ)] = h ◦ ϑ(д ) = h · Θ(д). ∈P † i=1 pi d q ∈P K[ ] i d Note that, since the representation ρ on x preserves degree, Last equality stems from a and b being different rows in Q. deghi , deghj ⇒ Θij (д) = 0, ∀д ∈ G. □ [ ] ∈ K[ ] × × Proposition 5.1. Consider h = h1,..., hℓ x d1 ... K[ ] ◦ ( −1) · ( ) ∈ x dl and assume that h ϑ д = h Θ д , for all д G. For any d ∈ N, the map ψd,h is τ − ρ equivariant for the representation τ on 5 SYMMETRY REDUCTION −1 K[x] − × ... × K[x] − defined by τ (д)(f) = Θ(д)· f ◦ ϑ(д ). The symmetries we deal with are given by the linear action of a d d1 d dℓ n −1 −1 finite group G on K . It is thus given by a representation ϑ of G on Proof. (ρ(д) ◦ ψd,h)(f) = ρ(д)(h · f) = h ◦ ϑ(д )· f ◦ ϑ(д ) = n [ ] −1 K . It induces a representation ρ of G on K x given by h · Θ(д)· f ◦ ϑ(д ) = (ψh ◦ τ (д)) (f). □ ρ(д)p(x) = p(ϑ(д−1)x). (9) It also induces a linear representation on the space of linear forms, By application of [9, Theorem 2.5], the matrix of ψd,h is block diagonal in symmetry adapted bases of K[x] × ... × K[x] the dual representation of ρ : d−d1 d−dℓ and K[x] . Yet, in the algorithm to compute symmetry adapted ∗ −1 ∗ d ρ (д)λ(p) = λ(ρ(д )p), p ∈ K[x] and λ ∈ K[x] . (10) H-basis, the set H increases with d at each iteration and τ changes We shall deal with an invariant subspace Λ of K[x]∗. Hence the accordingly. We proceed to discuss how to hasten the computation ∗ K[ ] × × restriction of ρ to Λ is a linear representation of G in Λ. of a symmetry adapted basis of the evolving space x d−d1 ... [ ] In the above Algorithm 2, to compute an H-basis of I = kerw, K x d−dℓ . we use the Vandermonde and Macaulay matrices. We showed in The set H = H 1 ∪ ... H N that we shall build, degree by degree, [33, Section 4.2] how the Vandermonde matrix can be block diag- is actually a symmetry adapted basis. In particular, for 1 ≤ i ≤ N , onalized using appropriate symmetry adapted bases of K[x] and Hi spans the isotypic component associated to the irreducible rep- Λ. We show here how to obtain such a block diagonalization on resentation ρi . If the multiplicity of the latter, in the span of H, is 6 i h iT ℓi then the cardinality of H is ℓi ni . The matrices of the represen- Once we have in hand H0 = h1 ,..., h1 ,..., hN and a 11 1n1 cN nN θ Θ(д) = (R (д) ⊗ I |i = ... N ) tation in this basis are diag i ℓi 1 . symmetry adapted basis for Λ↓, we compute H by interpolation. i Assume H is determined by hi, ,..., h , of respective de- 0 θ 1 i,ℓi Since H ∈ K[x] , by [33, Proposition 3.5], its interpolant in Λ↓ is grees d ,...,d . In other words, for 1 ≤ l ≤ ℓ , ϑ i,1 i,ℓi i also ϑ − θ equivariant. Therefore h = h , π (h ),..., π (h ) h iT i,l i,l i,21 i,l i,ni 1 i,l H = h1 − hg1 ,..., h1 − h1 ,..., hN − hŸN ∈ K[x]θ . 11 11 1n1 1n1 cN nN cN nN ϑ ◦ ( −1) · ( ) is such that hi,l ϑ д = hi,l Ri д . Hence the related product The set H of its component is thus a symmetry adapted basis. The subspace K[x]ni is invariant under τ . The symmetry adapted d−di,l correctness and termination of Algorithm 3 follow from the same bases for all these subspaces can be combined into a symmetry arguments exposed for Algorithm 2. Note that both Macaulay and  n1 ÍN adapted basis for the whole product space K[x]d × K[x]d × Vandermonde matrices split in ni blocks. Assuming that the 1,1 1,ℓ1 i=1  nN blocks are equally distributed and thanks to [37, Proposition 5] we ... × K[ ] × K[ ] K[ ]ni x dN ,1 x d1,ℓ . Note that the components x e Pi N i 0 W ( )( ) ( )· ◦ ( −1) M (H ) ≈ Li ≈ with representation τi,e defined by τi,e д f = Ri д f ϑ д can approximate the dimensions of the blocks by M(H0) P WL are bound to reappear several times in the overall algorithm of next 1 . Therefore depending on the size of G the dimensions of the τ |G | section. Hence the symmetry adapted bases for the evolving can matrices to deal with in Algorithm 3 can be considerably reduced. be computed dynamically. Algorithm 3 6 CONSTRUCTING SYMMETRY ADAPTED i,1 H-BASIS Input: - L a s.a.b of Λ (r = |L | = dim (Λ), ri = L ) - P an orthonormal graded s.a.b of K[x]≤r In this section we show, when the space Λ is invariant, an orthog- ni - Mi a graded s.a.b of K[x]≤r , 1 ≤ i ≤ N onal equivariant H-basis H can be computed. In this setting, we Output: - H an orthogonal equivariant H-basis for I := ker Λ exploit the symmetries of Λ to build H. A robust and symmetry - PΛ a s.a.b of the least interpolation space for Λ. 0 adapted version of Algorithm 2 is presented. The block diagonal 1: H ← {}, PΛ ← {} structure of the Vandermonde and Macaulay matrices allow to re- 2: d ← 0 duce the size of the matrices to deal with. The H-basis obtained as 3: L≤0 ← {}, L≥0 ← L the output of Algorithm 3 inherits the symmetries of Λ. 4: while L≥d , {} do to Li,1 ∅ 5: for i = 1 N such that ≥d , do Proposition 6.1. Let I = ∩ ker λ and d ∈ N. If Λ is invariant,   i,1 i,1 λ ∈Λ R , P P   6: Q · d i = W d ▷ QR-decomposition of W d then so are I, I0, I0 , V I0 . Also, if H is an orthogonal H-basis 0 Li,1 Li,1 d d

R3 Algorithm 3 is a symmetry adapted version of Algorithm 2. In Example 6.2. The subgroup of the orthogonal group that any iteration we compute H 0 as a symmetry adapted basis of the leaves the regular the cube invariant is commonly called Oh. It d has order 48 and 10 inequivalent irreducible representations whose orthogonal complement of V (H 0 ) in I0. d

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