UNCORRECTED PROOF Φ ⊗S Ψ Φψ 31 for L ≥ 0

YEUJC: 701 pp. 1–15 (col. ﬁg: NIL) PROD. TYPE: TEX

ARTICLE IN PRESS

European Journal of Combinatorics xx (xxxx) xxx–xxx www.elsevier.com/locate/ejc Spherical designs and ﬁnite group representations (some results of E. Bannai)

Pierre de la Harpe, Claude Pache

Section de Mathématiques, UniversitédeGenève, C.P. 240, CH-1211 Genève 24, Switzerland

Accepted 17 July 2003

1 Abstract

2 We reprove several results of Bannai concerning spherical t-designs and ﬁnite subgroups of 3 orthogonal groups. These include criteria in terms of harmonic representations of subgroups of O(n) n−1 4 for the corresponding orbits to be t-designs (t = 0, 1, 2, 3,...)in S . We also discuss a conjecture 5 of Bannai, dating from 1984, according to which t is bounded independently of the dimension n (for 6 n ≥ 3) for such designs. © 2003 Published by Elsevier Ltd.

7 MSC 1991: primary 05B30; secondary 20C15

8 Keywords: Spherical designs; Group representations; Homogeneous ﬁnite subgroups of orthogonal groups

9 1. Introduction

10 Given a dimension n ≥ 2 and an integer t ≥ 0, a spherical t-design in dimension n is a n−1 n 11 nonempty ﬁnite subset X of the unit sphere S of the Euclidean space R such that 1 φ(x) = φ(y) dµ(y) |X| Sn−1 12 x∈X (t) n−1 13 for all φ ∈ F (S ), the space of those real-valued continuous functions on the sphere n 14 which are restrictions of polynomial functions of degree at most t on R .Here|X| denotes n−1 15 the cardinality of X and µ the O(n)-invariant probability measure on S ,whereO(n) is n 16 the group of orthogonal transformations on R . 17 The term “spherical design” goes back to [9](seealso[11]and[29]). There is an 18 existence resultUNCORRECTED for all values of n and t [21](seealso[1 –3PROOF], and [31]), but explicit 19 examples are in general not straightforward to construct when n ≥ 3andt ≥ 2. However,

E-mail addresses: [email protected] (P. de la Harpe), [email protected] (C. Pache).

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n−1 for low values of t, results of Bannai provide spherical t-designs as orbits in S of finite 1 subgroups of O(n). The purpose of the present exposition is to prove some of these results 2 of Bannai in a way we find simpler than in the original articles; in particular, we avoid the 3 use of bases in spaces of harmonic polynomials. (As we were finishing this work, we found 4 out an exposition overlapping substantially with ours in Section 2 of [18].) 5 ( ) ≥ π(k) More precisely, let G be a subgroup of O n . For an integer k 0, let G denote the 6 (k) n n natural linear representation of G in the space H (R ) of real-valued polynomials on R 7 which are homogeneous of degree k and harmonic (see Section 2 below). We denote by 8 1G the unit representation of G,andρ ≮ σ means that the representation ρ of G is not a 9 subrepresentation of the representation σ of G. A finite subgroup G of O(n) is said to be 10 n−1 t-homogeneous if the orbit Gx0 of any point x0 ∈ S is a spherical t-design. 11

Theorem 1 (Bannai). Let G be a ﬁnite subgroup of O(n) and let s, t be positive integers. 12 ≮ π(k) ≤ ≤ (i) If 1G G for 1 k t, then G is t-homogeneous, and conversely. 13 ≥ π(k) ≤ ≤ ( ) (ii) If n 3 and if G is irreducible for 1 k s, then G is 2s -homogeneous. 14 ≥ π(k) ≤ ≤ π(s) ≮ π(s+1) (iii) If n 3,if G is irreducible for 1 k s, and if G G ,thenGis 15 (2s + 1)-homogeneous. 16 n−1 (iv) If there exists one orbit of G on S which is a spherical (2t)-design, then G is 17 t-homogeneous. 18

Moreover, for each integer n ≥ 3, there exists an integer tmax(n) such that, whenever some 19 ﬁnite subgroup of O(n) is t-homogeneous, then t ≤ tmax(n). 20

Claim (i) is essentially a reformulation of the deﬁnitions (it appears as Theorem 6.1 21 in [11]). Claims (ii) and (iii) appear in [4]and[5], with a slightly more restrictive 22 hypothesis. (In particular, it was observed in [11] that the absolute irreducibility of 23 π(k) G , assumed by Bannai, can be replaced by irreducibility; also, in (iii), the hypothesis 24 π(s) ≮ πs+1 G G is a weakening of the corresponding hypothesis by Bannai.) Claim (iv) and 25 the bound t ≤ tmax(n) appear in [6]and[7]. With appropriate deﬁnitions, claims (i)–(iv) 26 carry over to compact subgroups of O(n). 27 The converses of claims (ii) and (iii) do not hold, and the claims themselves do not hold 28 for n = 2 (see below, the end of Section 2). As the group of (t + 1)-roots of unity is a 29 1 spherical t-design in S for each t ≥ 0, the last claim in the theorem does not hold for 30 n = 2. 31 (After submission of this paper, C. Pache has found that, in claims (ii) and (iii) of 32 π(s) π(k) Theorem 1, it is enough to assume that G is irreducible, instead of assuming that G is 33 irreducible for 1 ≤ k ≤ s. See the Appendix below.) 34

Proposition 2 (Bannai). Let H be a ﬁnite subgroup of O(n), let X be a spherical (2s)- 35 n−1 design in S which is H-invariant, and let λ denote the permutation representation of H 36 λ s π(k) deﬁned by its action on X. Then contains k=0 H as a subrepresentation. 37

This is Theorem 2 in [6]. Claims (i)–(iii) of Theorem 1 are proved in Section 2.The 38 other claims ofUNCORRECTEDTheorem 1 and Proposition 2 are proved in Section PROOF 3. 39 Theorem 1 can be illustrated by numerous examples. In particular, consider in O(3) the 40 subgroup G(T ) (resp. G(C), G(D)) of orthogonal symmetries of a regular tetrahedron 41 YEUJC: 701 ARTICLE IN PRESS

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1 (resp. cube, dodecahedron). Then G(T ) is 2-homogeneous and has orbits which are 2 spherical 3-designs and 5-designs; G(C) is 3-homogeneous; G(D) is 5-homogeneous and 3 has orbits which are spherical 9-designs; see [12]. The group of automorphisms of a Leech 4 lattice, which is a ﬁnite subgroup of O(24), is 11-homogeneous and has orbits which are 5 spherical 15-designs (see Example 8.5 in [9], as well as Section 7 in [12]). There are other 6 examples in [22]and[23]. Constructions involving ﬁnite subgroups of O(n) in the closely 7 related subject of “spherical designs with weights” (better known as “cubature formulas on 8 spheres”) go back at least to [25].

9 We mention a few more examples in Section 4, which among other things makes precise 10 the following statement (which rules out subgroups of O(2)).

11 Proposition 3. There exists an inﬁnite family of 7-homogeneous groups, and there exist 12 11-homogeneous groups.

13 There is a conjecture of Bannai, according to which the last statement of Theorem 1 14 holds with a bound tmax independent of the dimension n, but we have not been able to 15 make progress on this. At the end of the paper, we formulate questions related to Bannai’s 16 conjecture.

17 2. Sufﬁcent conditions for t-homogeneity

18 Consider n ≥ 2andt ≥ 0 as in the Introduction, a ﬁnite subgroup G of O(n), a point n−1 19 x0 in the unit sphere S , and the orbit X = Gx0.Wehave 1 1 φ(x) = φ(gx ) |X| |G| 0 20 x∈X g∈G

n−1 21 for all functions φ on S .IfG is more generally a compact subgroup of O(n),wedeﬁne 22 an orbit Gx0 to be a spherical t-design if

φ(gx0) dg = φ(y) dµ(y) (1) 23 G Sn−1 φ ∈ F (t)(Sn−1) 24 for all ,wheredg denotes the normalized Haar measure on G. F (t)(Sn−1) ( (φ))( ) = φ( ) 25 Let p be the linear operator on deﬁned by p y G gy dg.Then (t) n−1 26 p is a projection onto the subspace of F (S ) of G-invariant functions, here written (t) n−1 G (0) n−1 27 F (S ) ; observe that this space contains the space F (S ) of constant functions, 28 which we identify with R. Thus, condition (1) reads

(p(φ))(x0) = φ(y) dµ(y) (2) 29 Sn−1 (t) n−1 30 for all φ ∈ F (S ).

31 Extending a previous deﬁnition, we say that a compact subgroup G of O(n) is ∈ Sn−1 32 t-homogeneousUNCORRECTEDif, for all x0 , the orbit Gx0 is a spherical PROOFt-design. 33 Given two vector spaces U and V , here over R, the space of linear mappings from U to 34 V is denoted below by L(U, V ). YEUJC: 701 ARTICLE IN PRESS

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G Whenever V is a space of functions on which a group G acts, we denote by V the 1 space of G-invariant functions. 2

Proposition 4. Let G be a compact subgroup of O(n). 3 (t) n−1 G (i) For t ≥ 0, the group G is t-homogeneous if and only if F (S ) = R. 4

In particular: 5 n (ii) the group G is 1-homogeneous if and only if G does not ﬁx any vector x = 0 in R ; 6 n (iii) the group G is 2-homogeneous if and only if the linear action of G on R is 7 irreducible. 8

Proof. Eq. (2)showsthatG is t-homogeneous if and only if p(φ) is a constant function 9 (t) n−1 for all φ ∈ F (S ), and this establishes (i). 10 (1) n−1 n (1) n−1 G For claim (ii), observe that F (S ) = R ⊕ L(R , R),sothatF (S ) = R if 11 n G n G and only if L(R , R) ={0}, if and only if (R ) ={0}. 12 For claim (iii), assume first that G is reducible; let V be a nontrivial G-invariant 13 n n subspace of R .Ifp denotes the orthogonal projection of R onto V and · | · the 14 n (2) n−1 G Euclidean scalar product, the function R x p(x) | p(x)∈R is in F (S ) 15 (2) n−1 G and is not constant, so that F (S ) = R. 16 (2) n−1 G Assume next that F (S ) contains a nonconstant function φ. If the odd part 17 1 (φ( ) − φ(− )) x 2 x x is not zero, G is reducible by (ii). We may therefore assume 18 n without loss of generality that φ : R R is a homogeneous polynomial of degree 2. Let 19 n β be the symmetric bilinear form defined on R by β(x, y) = φ(x + y) − φ(x) − φ(y) 20 n n n and let B ∈ L(R , R ) be the operator defined by β(x, y) = x | B(y) for all x, y ∈ R . 21 n Then B is self-adjoint, commutes with all elements of G, and any eigenspace of B in R is 22 G-invariant. Moreover, since φ is not constant, B is not a scalar multiple of the identity, 23 n thus B has a nontrivial eigenspace, and the action of G on R is reducible. (This implication 24 is a particular case of one in Theorem 7,provenbelow.) 25

Let us now review some classical facts on spherical representations of O(n).For 26 (k) n n k ≥ 0, let P (R ) denote the space of real-valued polynomial functions on R which 27 are homogeneous of degree k,andlet 28 2 2 ( ) ( ) ∂ φ ∂ φ H k (Rn) = φ ∈ P k (Rn) +···+ = 0 ∂ 2 ∂ 2 x1 xn 29 denote the space of harmonic polynomials of degree k. We will identify these spaces with 30 n−1 spaces of continuous functions on S . 31 Each of these spaces is also O(n)-invariant for the natural action. More precisely, for 32 (k) (k) n (k) k ≥ 0, the linear representation π of O(n) in H (R ) is deﬁned by (π (g)φ)(x) = 33 −1 (k) (k) n φ(g x); we denote by πC the complexiﬁed representation, in the space H (R ) ⊗R C. 34 The two following results are classical (see e.g. [26]or[30]). 35 (i) We have O(n)-invariant direct sums 36 F (UNCORRECTEDt)(Sn−1) = P(t)(Rn) ⊕ P(t−1)(Rn) = H(k )(PROOFRn) 37

for all t ≥ 0. 38 YEUJC: 701 ARTICLE IN PRESS

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(k) 1 (ii) The representations πC , k ≥ 0, are pairwise inequivalent irreducible complex (k) 2 representations of O(n). A fortiori, the π are pairwise inequivalent irreducible 3 real representations of O(n).

( ) π(k) π(k) 4 For any subgroup G of O n , we denote by G the restriction of to G.

5 Proof of Theorem 1(i). This follows from Proposition 4(i) and from the direct sum 6 decomposition in (i) above. (k) n 7 For ψ1,ψ2 ∈ P (R ), we have a differential operator with constant coefﬁcients n 8 ψ2(∂/∂x), and it happens that ψ2(∂/∂x)ψ1, a priori a function on R , is in fact a 9 constant function; moreover, if [ψ1 | ψ2] denotes the value of this constant function, the (k) n 10 assignment (ψ1,ψ2) [ψ1 | ψ2] deﬁnes a scalar product on P (R ), and therefore also (k) n (k) 11 on H (R ) by restriction. The representation π of O(n) is orthogonal for this scalar (k) (k) (k) n 12 product: [π (g)ψ1 | π (g)ψ2]=[ψ1 | ψ2] for all g ∈ O(n) and ψ1,ψ2 ∈ H (R ). 13 For l, m ≥ 0, we have a linear mapping H(l+m)(Rn) L(H(l)(Rn), H(m)(Rn)) µ : l,m φ ψ ψ ∂ φ . (3) 14 ∂x

15 Lemma 5. For all l , m ≥ 0, the mapping µl,m deﬁned by (3) is injective. sa (l) n (l) n 16 In the case l = m, the image of µl,l is inside the space L (H (R ), H (R )) of (l) n 17 operators on H (R ) which are self-adjoint with respect to the scalar product [· | ·]. ω ∈ P(2)(Rn) ω( ) = 2 +···+ 2 18 Proof. Denote by the function deﬁned by x x1 xn .Itisa (k) n 19 classical fact that any α ∈ P (R ) can be written in a unique way as α = θα + ραω (k) n (k−2) n 20 with θα ∈ H (R ) and ρα ∈ P (R ),sothatα θα is the orthogonal projection (k) n (k) n 21 from P (R ) onto H (R ) with respect to [· | ·]. Since the pointwise multiplication (l) n (m) n (l+m) n 22 P (R ) ⊗ P (R ) ψ ⊗ χ ψχ ∈ P (R ) is clearly onto, it follows that the (l) n (m) n (l+m) n 23 linear mapping H (R ) ⊗ H (R ) ψ ⊗ χ θψχ ∈ H (R ) is also onto. (l+m) n (l) n (m) n 24 Choose now φ ∈ H (R ).Forallψ ∈ H (R ) and χ ∈ H (R ),wehave

∂ ∂ ∂ ∂ ∂ [µl,m (φ)ψ | χ]=χ ψ φ = θχψ φ + ρχψ ω φ 25 ∂ ∂ ∂ ∂ ∂ x x x x x ∂ = θχψ φ =[φ | θχψ] 26 ∂x

27 since ω(∂/∂x)φ = 0. In particular, if φ ∈ Ker(µl,m ),then[φ | θ]=0forall (l+m) n 28 θ ∈ H (R ), and therefore φ = 0. Thus µl,m is injective. (2l) n (l) n 29 Assume now that l = m.Forφ ∈ H (R ) and ψ1,ψ2 ∈ H (R ),wehave

∂ ∂ [µl,l (φ)ψ1 | ψ2]=ψ2 ψ1 φ =[µl,l (φ)ψ2 | ψ1] 30 ∂x ∂x

31 and the operator µl,l (φ) is self-adjoint.

(l,m) (l) n (m) n 32 The naturalUNCORRECTED representation π of O(n) on L(H (R ), H PROOF(R )) is given by (l,m) (m) (l) −1 33 π (g)λ = π (g) ◦ λ ◦ π (g ) YEUJC: 701 ARTICLE IN PRESS

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(l) n (m) n for λ ∈ L(H (R ), H (R )),andg ∈ O(n). Observe that the application µl,m of (3) 1 (l+m) (l,m) is O(n)-equivariant for π and π . Though it is not used below, it can also be 2 (l,m) (l) (m) observed that π is equivalent to the tensor product of the representations π and π 3 (l) (note that π , being orthogonal, is equivalent to its own contragredient). 4 sa (l) n (l) n (l,l) In the case l = m, the space L (H (R ), H (R )) is π (O(n))-invariant. Let 5 (l) n (l) n L0(H (R ), H (R )) denote the space of endomorphisms of trace zero. The space 6 Lsa(H(l)(Rn), H(l)(Rn)) = Lsa(H(l)(Rn), H(l)(Rn)) ∩ L (H(l)(Rn), H(l)(Rn)) 0 0 7 (l,l) is also π (O(n))-invariant. 8

Lemma 6. For l ≥ 1, the image of µl,l is inside the space of self-adjoint operators of trace 9 (l) n zero on H (R ), so that we have a mapping 10 µ : H(2l)(Rn) Lsa(H(l)(Rn), H(l)(Rn)) l,l 0 11 which is O(n)-equivariant for the natural representations. 12

Proof. Consider the sequence 13 (2l) n sa (l) n (l) n H (R ) L (H (R ), H (R )) R, 14 where the ﬁrst mapping is µl,l and the second is the trace. The two mappings are O(n)- 15 (2l) n (2l) n equivariant, dimR(H (R )) > 1, and H (R ) is O(n)-irreducible; it follows that the 16 composition of these two mappings is zero. 17

Theorem 7. Let G be a compact subgroup of O(n) and let t ≥ 0 be an integer. Assume 18 that each integer k such that 1 ≤ k ≤ t is given as a sum k = l +m of nonnegative integers, 19 and that 20

(l) n (m) n G L(H (R ), H (R )) ={0} in case l = m, 21 π(l) = G is irreducible in case l m. 22

Then G is t-homogeneous. 23

(k) n G Proof. By Theorem 1(i), it is enough to show that H (R ) ={0} whenever 1 ≤ k ≤ t. 24

In the case k = l + m with l = m, the existence of the O(n)-equivariant 25 (k) n G mapping (3), which is injective by Lemma 5, implies that H (R ) embeds in 26 (l) n (m) n G L(H (R ), H (R )) , and is therefore {0}. 27 (k) n G Similarly, in the case k = l + l with l ≥ 1, the space H (R ) embeds by Lemma 6 28 in 29 Lsa(H(l)(Rn), H(l)(Rn))G ≈{ }, 0 0 30 where the last isomorphism follows from Schur’s lemma. 31

Here is theUNCORRECTED form of Schur’s lemma used in the previous argument. PROOF Let V be a ﬁnite- 32 dimensional real vector space given together with a Euclidean scalar product and let π 33 Lsa( , )G be an orthogonal representation of some group G in V .Let 0 V V denote the space 34 YEUJC: 701 ARTICLE IN PRESS

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1 of those operators A : V V which are self-adjoint, of trace 0, and such that π( ) = π( ) ∈ π Lsa( , )G ={ } 2 A g g A for all g G.If is irreducible, then 0 V V 0 . ( j) 3 Proof of claims (ii) and (iii) in Theorem 1. Observe that two representations π and G π( j ) = 4 G with j j are never equivalent, since their dimensions are distinct. Apply now 5 Theorem 7 with k = l + l if k = 2l is even and k = l + (l + 1) if k = 2l + 1 is odd.

6 Consider an integer m ≥ 2 and a dihedral subgroup G of order 2m if O(2). On the one 7 hand, G is (m − 1)-homogenous and is not m-homogenous. On the other hand, the largest π(k) ∈{ ,..., } = ( / ) − 8 integer s such that G is irreducible for all k 1 s is s m 2 1ism if even 9 and s = m − 1ifm is odd. It follows that neither claim (ii) nor (iii) of Theorem 1 carry 10 over to the case n = 2.

11 On the converses of claims (ii) and (iii) in Theorem 1

2n−1 n 2n 12 The subgroup U(n) of O(2n) is transitive on the unit sphere S of C = R ,and ≥ π(2) 13 therefore is t-homogeneous for all t 0. However the representation U(n) is reducible. n 14 Indeed, let |C denote the scalar product on C , so that the Euclidean scalar product 2n n 15 on R = C is given by x | y=R( x | yC).ThereisaU(n)-invariant decomposition (2) 2n (2,0) n (1,1) n (0,2) n 16 H (R ) ⊗R C = HC (C ) ⊕ HC (C ) ⊕ HC (C )

(p,q) n n 1 17 where HC (C ) is the space of harmonic polynomial functions C C which are 18 homogeneous of degree p in z1,...,zn and homogeneous of degree q in z1,...,zn.We 19 have a U(n)-invariant direct sum

H(2)(R2n) = H(1,1)(Cn) ∩ H(2)(R2n) 20 C

H(2,0)(Cn) ⊕ H(0,2)(Cn) ∩ H(2)(R2n) . 21 C C

22 The ﬁrst factor contains functions of the form

23 (x1,...,x2n) α | zC α | zC

24 and the second factor contains functions of the form 2 2 25 (x1,...,x2n) α | zC + α | zC 2n 26 with α ∈ R . 27 There is in [6] an example of a ﬁnite group showing that the converses of claims (ii) and 28 (iii) in Theorem 1 do not hold.

1 2n n It is convenient to use on R = C not only the canonical coordinates (x1,...,x2n), but also coordinates (z1,...,zn, z1,...,zn), with z j = x j + ixn+ j and z j = x j − ixn+ j for 1 ≤ j ≤ n. A smooth function ∂2 (p,q) n φ (R-valued or C-valued) is then harmonic if ≤ ≤ φ(z, z) = 0. Let P (C ) denote the space 1 j n ∂z j ∂z j C R2n C ,..., of polynomial functionsUNCORRECTEDwhich are homogeneous of degree p PROOFin z1 zn and homogeneous (p,q) n of degree q in z1,...,zn.ThenHC (C ) is the kernel of the Laplacian viewed as a linear mapping (p,q) n (p−1,q−1) n PC (C ) PC (C ). YEUJC: 701 ARTICLE IN PRESS

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3. Upper bounds on homogeneity 1

H\X If H is a group which acts (here on the left) on a set X, we denote by R the space 2 of real-valued functions on the orbit space H \X. As before, n ≥ 2ands ≥ 0aregiven 3 integers. 4 n−1 Lemma 8. Let X ⊂ S be a spherical (2s)-design. Then the linear mapping 5 F (s)(Sn−1) RX EvX : φ (x φ(x)) 6 is injective. If X is moreover invariant by some ﬁnite subgroup H of O(n),thenEvX is 7 H-equivariant, so that in particular the mapping 8 F (s)(Sn−1)H RH\X

φ (Hx φ(Hx)) 9 is also injective. 10 (s) n−1 X Proof. For φ in the kernel of the mapping F (S ) R ,wehave 11 1 φ2(y) dµ(y) = φ2(x) = 0 Sn−1 |X| x∈X 12 and therefore φ = 0, so that EvX is injective. The other claims are straightforward to 13 check. 14

Though we will not need it here, observe that an immediate consequence of Lemma 8 15 is the well-known inequality 16

+ − + − (s) n−1 n s 1 n s 2 |X|≥dimR(F (S )) = + s s − 1 17 for any spherical (2s)-design. A second observation is that, in the case −X = X is 18 an antipodal spherical (2s + 1)-design, an analogous lemma shows that the restriction 19 (s) n−1 Y mapping P (S ) R is injective, where Y ⊂ X is any subset such that X = 20 Y ∪ (−Y ) and Y ∩ (−Y ) =∅; this in turn implies that 21

+ − (s) n n s 1 |X|=2|Y |≥2dimR(P (R )) = 2 s 22 for any antipodal spherical (2s + 1)-design (see Theorem 5.11 of [9] for a proof not using 23 the hypothesis −X = X). 24

Proof of Theorem 1(iv) and Proposition 2. If H is a ﬁnite group which is transitive 25 (s) n−1 H on a spherical (2s)-design X, Lemma 8 implies that F (S ) = R. Thus H is 26 s-homogeneous by part (i) of Theorem 1. 27

Proposition 2 follows from the H -equivariance in Lemma 8, and from the fact that the 28 (s) n−1 s (k) n direct sum F (S ) = H (R ) is H -invariant (indeed O(n)-invariant). 29 UNCORRECTEDk=0 PROOF Proof of the last claim of Theorem 1. Let c(n) be a bound for the theorem of Jordan on 30 normal Abelian subgroups of ﬁnite linear groups. Let G be a ﬁnite subgroup of O(n). 31 YEUJC: 701 ARTICLE IN PRESS

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1 We can choose an Abelian subgroup A of G of index at most c(n). Consider a point n−1 2 x0 ∈ S , its orbit X = Gx0, and assume that X is a spherical t-design for some t ≥ 0. 3 Observe that X is invariant by A and that |A\X|≤c(n). Thus

([t/2]) n−1 A 4 dimR(F (S ) ) ≤|A\X|≤c(n)

5 by Lemma 8 (here [t/2] denotes the integer part of t/2). n 6 Assume now that n ≥ 3. As the representation of the Abelian group A on R is (2) n A 7 reducible, there exists a polynomial f ∈ H (R ) which is not zero. More precisely, 8 in appropriate coordinates, we can set ( ,..., ) = ( − )( 2 + 2) − ( 2 +···+ 2). 9 f x1 xn n 2 x1 x2 2 x3 xn n−1 10 As f deﬁnes a continuous function on S which is not constant (and therefore which 11 takes inﬁnitely many values), the only polynomial expression of the form c0 + c1 f + 2 k n−1 12 c2 f +···+ck f whichiszeroonS is that with c0 = c1 = ··· = ck = 0. In other 2 k (2k) n−1 A 13 words, the functions 1, f, f ,..., f are linearly independent, and in F (S ) .In 14 particular, we have t ([t/2]) n−1 A + 1 ≤ dimR(F (S ) ) ≤ c(n) 15 4 16 and t ≤ 4c(n) − 1.

17 Remark. In the proof above, we can set √ √ 2n2 2n2 18 c(n) = ( 8n + 1) − ( 8n − 1) ;

19 see, e.g. [10]. (There in [32] a discussion of the bound in Jordan’s theorem using the 20 classiﬁcation of ﬁnite simple groups.) Observe however that this bound holds for the index 21 of a normal Abelian subgroup of G, whereas we have only used that A is an Abelian 22 subgroup of G.

23 4. Examples of t-homogeneous groups for 3 ≤ t ≤ 11

( ) (k) H(k)(Rn)G 24 Let G be a ﬁnite subgroup of O n . The dimensions hG of the spaces of 25 G-invariant harmonic polynomials can be computed from the adapted Molien–Poincar´e 26 series

∞ 2 ( ) 1 1 − T h k T k = G |G| det(1 − gT) 27 k=0 g∈G

28 which is an equality between formal power series; see No. V.5.3 in [8], or [24]. Thus, at 29 least in principle, the maximal t for which G < O(n) is t-homogeneous can be found Rn H(k)(Rn) ≥ 30 out with computationsUNCORRECTED involving the action of G on only, notPROOF on for k 2. 31 (Actual computations are however known to be “in general” as complicated as possible: 32 see [15].) YEUJC: 701 ARTICLE IN PRESS

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In the case of an irreducible finite group W generated by l ≥ 2 reflections, this series is 1 of the form 2 ∞ l (k) k = 1 h T + G 1 − T mi 1 k=0 i=2 3 where the increasing sequence m1 = 1 < m2 ≤···≤ml is that of the Coxeter exponents 4 of the Coxeter group W;seeNo.V.6.2in[8] or Chapter 3 in [16]. It follows that W is 5 l−1 m2-homogeneous, and that some orbits of W on S are spherical m3-designs when l ≥ 3. 6 In particular, with standard notation for the types of finite Coxeter groups, we have the 7 following list. 8

W(Al ) ≈ Sym(l + 1) is 2-homogeneous for l ≥ 2, and m3 = 3forl ≥ 3. 9 l W(Bl ) ≈ (Z/2Z) Sym(l) is 3-homogeneous for l ≥ 2, and m3 = 5forl ≥ 3. 10 l−1 W(Dl ) ≈ (Z/2Z) Sym(l) is 3-homogeneous for l ≥ 4, and m3 = min{5, l − 1}. 11 ( p) ≈ (Z/ Z) (Z/ Z) ( − ) = 2 ≥ W I2 p 2 is p 1 -homogeneous for p 5and p 7. 12 W(E6) is 4-homogeneous, and m3 = 5. 13 W(E7), W(F4), W(H3) are 5-homogeneous, and m3 = 7, 7, 9 respectively. 14 W(E8) is 7-homogeneous, and m3 = 11. 15 W(H4) is 11-homogeneous, and m3 = 19. 16

Some root systems are example of exceptional orbits: those of types A2, D4,andE6 are 17 spherical 5-designs which are orbits of t-homogeneous groups for t = 2, 3, 4 respectively. 18 (Let us mention that infinite Coxeter groups are also relevant to spherical designs [14].) 19 There are a large number of pairs G < O(n) where n ≥ 3andG is a finite subgroup 20 ( ) π(1) π(2) {± } of O n with G and G irreducible, such that the group generated by G and 1 21 is 5-homogeneous. Cases with G quasi-simple (i.e. simple modulo its centre) have been 22 classifiedin[18](seealso[19]). In particular, there exist several infinite families of 23 such examples. There exist also an infinite family and some isolated examples of pairs 24 G < O(n), with n ≥ 3, such that the finite group G is 7-homogeneous (see [29], in 25 particular Remark 18.10); the groups of the infinite family are the automorphism groups 26 of the so-called Barnes–Wall lattices. We know two examples of 11-homogeneous groups, 27 which are W(H4)

(Q1) Do there exist inﬁnitely many pairs G < O(n), with n ≥ 3, such that the ﬁnite group 33 G is t-homogeneous for some t > 7? 34

(Q2) Do there exist examples G < O(n), with n ≥ 3, such that the ﬁnite group G is 35 > t-homogeneousUNCORRECTED for some t 11? PROOF 36

2 = 6 Also for p 6, except that I2 is then conventionally written G2. YEUJC: 701 ARTICLE IN PRESS

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1 5. Analogous questions for lattice designs Rn ( ) = { | : ∈ , = } 2 Let√L be a lattice in with minimal norm N L min x x x L x 0 .Set 3 ρ = N(L) and let n−1 4 X L ={x ∈ S : ρx ∈ L}

5 be its set of normalized short vectors. A natural question to ask is: for which t is the set X L 6 a spherical t-design? We report below some information communicated to us by Venkov.

7 The theory of extreme lattices (namely of lattice sphere packing of highest density) 8 motivates the study of strongly eutactic and strongly perfect lattices, defined as those for 9 which X L is respectively a spherical 3-design and a spherical 5-design. (A lattice is extreme 10 if the density of the corresponding lattice sphere packing is a local maximum in the space 11 of all lattices of the same rank. It is a theorem of Vorono¨ı that a lattice is extreme if and 12 only if it is “eutactic” and “perfect”; see for example Chapter 4 of [20].) The number of 13 similarity classes of strongly perfect lattice is finite in any dimension, and it is conjectured 14 that there exists at least one in any large enough dimension. There are exactly 10 similarity 15 classes of strongly perfect lattices in dimensions ≤11, traditionally denoted by A1, A2, D4, ∗ ∗ ∗ 16 E6, E6 , E7, E7 , E8, K10,andK10. For the 29 known similarity classes of strongly perfect 17 lattices in dimensions 12 ≤ n ≤ 23, see Tables 19.1 and 19.2 in [29]. n 18 Consider a unimodular even integral lattice L < R such that x | x≥4forall 19 x ∈ L, x = 0. Then X L is always a spherical 5-design. If n = 24, then L is a Leech lattice, 20 and the spherical 11-design X L appears already above as the orbit of the Conway group ·0. 7 21 Otherwise, n ≥ 32. For n = 32, it is known that there are more than 10 nonsimilar 22 lattices of this kind [17]; the group Aut(L) is far from being transitive on X L except in a 23 very small number of cases, indeed there are such L with Aut(L) ={±1}; in particular, 31 24 there is a large number of spherical 5-designs in S which are not simply related to orbits 25 of finite subgroups of O(32). 26 We have already mentioned the infinite family of Barnes–Wall lattices, for which X L 27 is a spherical 7-design. There are known lattices for which X L is a spherical 11-design, 28 such as the Leech latttice in dimension n = 24, and three lattices in dimension n = 48; for 29 some of this, see [28]. It is an open problem to know if there are such lattices in dimension 30 n = 72, and it is conjectured that there are none unless n ≡ 0(mod 24).

31 Conversely, it is also possible to define lattices in terms of appropriate spherical designs. n−1 n 32 More precisely, if X ⊂ S is a finite subset linearly generating R such that x | y∈Q n 33 for all x, y ∈ X,andif XZ denotes the additive subgroup of R generated by X, then the n 34 appropriate homothetic image L X = ρ XZ is an integral lattice in R . In particular, let G n−1 35 be a finite subgroup of O(n) ∩ GL(n, Q),letx0 ∈ S ,andletX = Gx0.ThenL X is a n−1 36 G-invariant integral lattice. If G is t-homogeneous, then the homothetic images in S of 37 all nonempty layers {x ∈ L | x | x=r}, r > 0, are spherical t-designs. n 38 Consider a lattice L < R .Letx0 ∈ L be primitive (namely x0 = 0, and Rx0 ∩ L = Z 2 = | ={ ∈ Sn−1 | ∈ } 39 x0). Set r0 x0 x0 and X x r0x L .Thenr0 L X is obviously 40 contained in L, and the inclusion is in general strict. There are a few exceptional cases = 41 such that r0 L XUNCORRECTEDL for all choices of x0. For example, this isPROOF so for the root lattice of 248 42 type E8, for the Leech lattice, and for the Thompson–Smith lattice T < R ,ofwhichthe 43 automorphism group modulo its centre {±1} is the Thompson group (the sporadic simple YEUJC: 701 ARTICLE IN PRESS

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16 ﬁnite group of order near 9 × 10 ). All layers of T are known to be spherical 7-designs. 1 See [27]andSection10in[13]. 2 It seems therefore natural to formulate the two following questions, analogous to those 3 of Section 4. 4

n (Q3) Do there exist inﬁnitely many similarity classes of lattices L < R such that X L is a 5 spherical t-design, for t > 7? 6 n (Q4) Do there exist examples of lattices L < R such that X L is a spherical t-design, for 7 t > 11? 8 = = ( ) = There is a lattice L K21 (notation of [29]) in dimension n 21 such that N L 4, 9 for which the set of short vectors {x ∈ L | x | x=4} provides a spherical 5-design, but 10 such that some layers {x ∈ L | x | x=r} for r > 4 provide only spherical 3-designs. 11 ∗ (L is the only known strongly perfect lattice of which the dual K21 is not strongly perfect.) 12 Thus, it also makes sense to ask questions similar to (Q3) and (Q4), involving all layers of 13 the lattices, rather than just the layer of short vectors. 14

Acknowledgements 15

The authors acknowledge support from the Swiss National Science Foundation. We are 16 grateful to B. Venkov for stimulating tutorials on spherical designs and lattices, and to the 17 Swiss National Science Foundation who made possible his visit to Geneva. 18

Appendix 19

The purpose of this Appendix is to present another proof of claims (ii) and (iii) of 20 Theorem 1, which shows the following sharper result. 21

Theorem A.1. Let G be a ﬁnite subgroup of O(n) and let s be a positive integer. 22

≥ π(s) ( ) (ii) If n 3 and if G is irreducible, then G is 2s -homogeneous. 23 ≥ π(s) π(s) ≮ π(s+1) ( + ) (iii) If n 3,if G is irreducible, and if G G ,thenGis 2s 1 -homogeneous. 24

The main ingredient of the proof below is a family (νl,m )l,m≥0 of mappings which can 25 be seen as a variation on the family (µl,m )l,m≥0 constructed just before Lemma 5.Fora 26 2 vector space V , we denote by Sym (V ) the symmetric square of V ;foru, v ∈ V ,we 27 2 denote by u ⊗s v the element 1/2(u ⊗ v + v ⊗ u) of Sym (V ).Wedeﬁne 28 H(l)(Rn) ⊗ H(m)(Rn) P(l+m)(Rn) νl,m : φ ⊗ ψ φψ 29 if l = m,and 30 2(H(l)(Rn)) P(2l)(Rn) ν : Sym l,l UNCORRECTED PROOF φ ⊗s ψ φψ 31 for l ≥ 0. 32 YEUJC: 701 ARTICLE IN PRESS

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(i) n 1 Lemma A.2. For n ≥ 3 and for each l, m ≥ 0, the image of νl,m contains H (R ) for 2 every i such that |l − m|≤i ≤ l + m and i ≡ l + m(mod 2).

3 Proof. It is known that, for each nonnegative integer k, there exists a unique polynomial (k) [k/2] (k) k−2i n 4 Q (T ) = q T ∈ R[T ] of degree k, such that, for every x ∈ R , the function i=0 i (k) (k)( ) := [k/2] (k) | i | k−2i | i H(k)(Rn) 5 Qx deﬁned by Qx y i=0 qi x x x y y y lies in ,and 6 such that (k)( ) ( ) µ( ) = ( ), ∀ ∈ Sn−1; Qx y f y d y f x x 7 S

n−1 (k) 8 see [9]. (For x ∈ S , observe that Qx is the unique homogeneous function of degree (k) (k) n−1 (k) 9 k such that Qx (y) = Q ( x | y) for every y ∈ S .) The polynomials Q (T ) are λ 10 related to the usual Gegenbauer (or ultraspherical) polynomials Ck by

( ) n + 2k − 2 ( − )/ Q k (T ) = C n 2 2(T ). 11 n − 2 k (l,m) 12 There exist constants qi such that (l)( ) (m)( ) = (l,m) (i)( ), Q T Q T qi Q T 13 i≥0 (l,m) = | − |≤ ≤ + ≡ + ( ) > 14 with qi 0 if and only if l m i l m and i l m mod 2 ,whenn 2; see n−1 15 for example [33, Formula (5.7)]. Let us choose a point e ∈ S . We conclude the proof 16 with the following lemma, applied to the group G = O(n),theimageW of νl,m , the vector v = (l) (m) v = (l,m) (i) 17 Qe Qe , and the components i qi Qe . = 18 Lemma A.3. Let V i∈I Vi be a ﬁnite direct sum of pairwise inequivalent irreducible v = 19 representations of a compact group G. Let W be a subrepresentation of V . Let v ∈ v ∈ ∈ v = ⊂ 20 i∈I i W with i Vi . Then, for every i I such that i 0, we have Vi W.

21 Proof. Let π : G → GL(V ) be a representation satisfying the hypotheses of the lemma, 22 and, for every i ∈ I,letχi be the character of the subrepresentation Vi . Since the Vi ’s are : → 23 pairwise inequivalent, the projection pi V Vi of kernel j=i V j can be written

pi = χi (g)π(g) dg. 24 G

25 It follows that pi W ⊂ W.Sowehavevi = pi v ∈ W ∩ Vi .SinceVi is irreducible, we have 26 Vi ⊂ W whenever vi = 0.

27 Lemma A.4. Let V and W be two real ﬁnite-dimensional orthogonal representations of a 28 group G. Then

G 29 (i) V and W are disjoint if and only if (V ⊗ W) ={0}; ( 2( )G ) = 30 (ii) V is irreducibleUNCORRECTED if and only if dimR Sym V 1. PROOF 31 Proof. We remark ﬁrst that a real ﬁnite-dimensional orthogonal representation is ∗ 32 equivalent to its contragredient, so V ⊗ W is equivalent to L(V, W) = V ⊗ W,and YEUJC: 701 ARTICLE IN PRESS

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2 sa Sym (V ) is equivalent to L (V ), the space of self-adjoint operators on V . Clearly, V and 1 G G W are disjoint if and only if L(V, W) ={0}, if and only if (V ⊗ W) ={0}.Also,V is 2 sa G 2 G irreducible if and only if L (V ) = R, if and only if dimR(Sym (V ) ) = 1. 3 π(s) Proof of Theorem A.1(ii) and (iii). Let us suppose that G is irreducible. We have to 4 (k) n G show that H (R ) ={0} for 1 ≤ k ≤ s. 5 2 (s) n By Lemma A.4,Sym(H (R )) is of dimension 1, and, by Lemma A.2, 6 s H(2 j)(Rn)G 2(H(s)(Rn))G H(0)(Rn)G = R j=0 is a subrepresentation of Sym .Since , 7 (2 j) n G we have H (R ) ={0} for 1 ≤ j ≤ s. 8 (k) n (l) n (s) Since n ≥ 3, if k < l,thendimR(H (R )) < dimR(H (R )). Therefore, if π is 9 (s) (s−1) (s) n irreducible, π and π are necessarily disjoint. Thus, by Lemma A.4, (H (R ) ⊗ 10 H(s−1)(Rn))G ={} s H(2 j−1)(Rn)G 0 .Now,byLemma A.2, j=1 is a subrepresentation 11 (s) n (s−1) n G (2 j−1) n G of (H (R ) ⊗ H (R )) . Therefore, H (R ) ={0} for 1 ≤ j ≤ s.This 12 terminates the proof of claim (ii). 13 Claim (iii) is proven by a similar argument. 14

Remark. For n = 3 and for every l ≥ 0, the mapping νl,l is a bijection, since the 15 dimensions of its source and of its target are the same. It follows that 16

( ) ( ) ( ) Sym2(H l (Rn)) and P 2l (Rn) = H 2i (Rn) 17 are equivalent representations of O(n). As a consequence, the converse of Theorem A.1(ii) 18 = < ( ) π(s) π(k) does hold for n 3. In particular, for G O n ,if G is irreducible, then G is 19 irreducible for every k ≤ s. 20 The statement 21 ( ) π(s) π(k) for G a ﬁnite (or a compact) subgroup of O n , if G is irreducible, then G is 22 irreducible for every k ≤ s 23 is true for n = 3 (see above), but is false for n = 2. We do not know whether it holds or 24 not for n ≥ 4. 25

References 26

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UNCORRECTED PROOF