DIRECT CONSTRUCTION OF SYMMETRY-BREAKING DIRECTIONS IN BIFURCATION PROBLEMS WITH SPHERICAL SYMMETRY

. We consider bifurcation problems in the presence of O (3) symmetry. Well known group-theoretic techniques enable the classiﬁcation of all maximal isotropy subgroups of O (3), with associated mode numbers (cid:96) ∈ N , leading to 1-dimensional ﬁxed-point subspaces of the (2 (cid:96) +1)-dimensional space of spherical harmonics. In each case the so-called equivariant branching lemma can then be used to establish the existence of a local branch of bifurcating so- lutions having the symmetry of the respective subgroup. To ﬁrst-order, such a branch is a precise linear combination of the 2 (cid:96) +1 spherical harmonics, which we call the bifurcation direction. Our work here is focused on the direct construction of these bifurcation directions, complementing the above-mentioned classiﬁcation. The approach is an application of a general method for constructing families of symmetric spherical harmonics, based on diﬀerentiating the fundamental solution of Laplace’s equation in R 3 .

subject to the constraint Here > 0 is a fixed small (material) parameter, σ = W , where W : R → R is a smooth 2-well potential, λ ∈ R is the bifurcation parameter, and λ+u represents the total value of the phase field on S 2 . Following the development in [4], we associate (1) with that of finding the zeros of a mapping F : R × X → Y ,viz., where the Banach spaces X := C 2,α (S 2 ) ∩ Y, Y := u ∈ C 0,α (S 2 ) : are equipped with the usual Hölder norms (0 < α ≤ 1). Its easy to see that u ≡ 0 satisfies (1) for all values of λ, i.e., F (λ, 0) ≡ 0. The rigorous linearization of (1) along the trivial line of solutions is then given by for h ∈ X, where A(λ) : X → Y is a Fredholm operator operator of index zero. Equation (4) admits nontrivial solutions if and only if λ ∈ R satisfies the characteristic equation − σ (λ)/ = ( + 1) for ∈ N.
In view of (4), observe that A(λ )[h] = 0 for all h ∈ V . The natural action of O(3) on functions in X (or Y ) is given by where Γ is a 3 × 3 orthogonal matrix, and x ∈ R 3 with |x| = 1. It easily follows from (1) that (3) is equivariant with respect to the natural action, viz., Let G ⊂ O(3) be a subgroup, and define the linear subspace Then (10) implies that F (λ, ·) : X G → Y G , where Y G is defined as in (11). From the classification in ( [3]; Thm. XIII.9.9), we are given the subgroups G ⊂ O(3) and corresponding mode numbers ∈ N such that (3) restricted to R × X G yields a standard 1-dimensional bifurcation problem, i.e., dim NullA( )| X G = 1, cf. (4). In this work we are concerned with finding the precise linear combination, such that Y * ∈ X G , where the coefficients, c m , − ≤ m ≤ , are to be determined. Let T (Γ) denote the (2 + 1) cf. [14]. Then (9), (11), (12) and (13) yield, where Of course it is enough to enforce the right side of (4) for the generators of G, suggesting a rather clumsy approach to "searching" for c ∈ R 2 +1 . More systematically, the average of T over the subgroup G yields a projection onto a 1-dimensional subspace of R 2 +1 , any non-zero element of which serves as c. However, this requires assembling the representation T explicitly, which is an inconvenient if not formidable task for even moderate values of ∈ N. In this work we present a direct and elegant alternative to the right side of (14), based on an old observation of Maxwell, i.e., that any spherical harmonic can be realized as a sequence of directional derivatives acting on on the fundamental solution of Laplace's equation in R 3 , viz., 1/r, where r 2 = x 2 + y 2 + z 2 . It was first recognized by Poole [11], and later by Hodgkinson [7], that complete families of spherical harmonics invariant with respect to the symmetry of a Platonic solid can be constructed by such a procedure. Later Meyer 1672 SANJAY DHARMAVARAM AND TIMOTHY J. HEALEY [10] presented a rigorous, systematic approach to that construction accounting for all possible subgroups of O(3). Here we merely specialize the approach [10] to the subspaces V , cf. (8) corresponding to any subgroup G ⊂ O(3). In particular, for the subgroups catalogued in [3], we can obtain any bifurcation mode Y * ∈ X G by successive differentiations. For subgroups outside of those in [3] the same method can be used to construct bases for Null A(λ )| X G .
The outline of this work is as follows: In Section 2 we consider the tetrahedral subgroup T and provide the details for constructing T-invariant spherical harmonics via directional derivatives of 1/r. In addition we provide a recipe for expressing these directional derivatives in terms of standard spherical harmonics. In Section 3 we consider all other finite subgroups of O(3). In Table 1, based on the work of Meyer [10], we summarizes the invariant spherical harmonic basis functions tabulated by their respective subgroups. We provide several examples in Section 4, mostly illustrating the utility of the method for constructing bifurcation directions, as described above. We pick one or two examples from each of the discrete subgroups listed in Theorem 9.9 of [3], constructing the bifurcation direction and illustrating its nodal pattern in each case.
2.1. Spherical harmonics as directional derivatives. Let us first recall that if y(x) is a harmonic function, i.e., ∆y(x) = 0 on R 3 , where ∆ is the Laplacian on R 3 , then y(x)| r=1 is a surface harmonic. Maxwell's insight on the connection between spherical harmonics and directional derivatives follows from the simple observation that 1/r is a harmonic function on R 3 − {0}, By taking the directional derivative of (15) in the direction a := (a x , a y , a z ) ∈ R 3 , we obtain ∆ D a (1/r) = 0, where D a (·) := [a x ∂ x + a y ∂ y + a z ∂ z ](·) is the directional derivative along a. Therefore, D a (1/r)| r=1 is a spherical harmonic. Similarly, by taking higher directional derivatives of (15) along a 1 , · · · a ∈ R 3 , ∈ N, we obtain Thus, D a1 D a2 · · · D a 1 r | r=1 is a spherical harmonic of order . The converse, that every spherical harmonic of order can be uniquely written as a product of directional derivatives of 1/r (up to reordering) is called Sylverster's theorem [13]. This relation between directional derivatives and spherical harmonics can be used to construct spherical harmonics that are invariant under the action of a subgroup of G ⊂ O(3), e.g., Y * ∈ X G , cf. (12).
The three dimensional representation corresponding to = 1 is precisely In view of (9), observe that γ(1/r) = (1/r), and since D a (1/r) = −a · x/r 3 , it follows that γ(D a (1/r)) = −(a · Γ T x)/r 3 = D Γa (1/r). More generally, we have It follows from (18) that by choosing directions a 1 , · · · , a such that we can construct spherical harmonics that are invariant under the action of G. For axi-symmetric spherical harmonics, the obvious choice for directional derivatives is along the axis of symmetry. For polyhedral subgroups of O(3) the vectors may be chosen to lie along symmetry directions of the corresponding regular polyhedron [10], [11].

2.2.
Constructing invariant spherical harmonics: Tetrahedral case. As an illustration of the utility of (19), we first consider the tetrahedral group T ⊂ O(3), viz., the proper-rotational symmetry group of a regular tetrahedron, to obtain Tinvariant spherical harmonics. For a tetrahedron oriented as depicted in Figure 1, the matrices generate the group. Consider the operator, where i, j, k are the cartesian basis vectors of R 3 (pointing along x, y and z axes, resp.). Using (18) and which, when evaluated at r = 1 and expressed in spherical coordinates results in where the numerical prefactor of Y 3,−2 in (25) results form the normalization constant, cf. (7). The function (25) (27) It is easy to verify (following (22) and (23)) that T 4 (1/r)| r=1 and T 6 (1/r)| r=1 are T-invariant spherical harmonics. The subscripts in (21), (26) and (27) remind the reader the order of derivatives, which is equivalent to the order of spherical harmonics.
Note that the operators (21), (26) and (27) can be mapped to polynomials by the formal identification: This connection between operators and polynomials is convenient, and we henceforth employ the following notation: Accordingly, (21), (26) and (27) now read (32) Since the choice of directional derivatives leading to a given spherical harmonic is not necessarily unique, multiple copies of the same set of vectors can be used to produce higher order spherical harmonics. For instance, T p 3 (1/r)| r=1 , T q 4 (1/r)| r=1 , T s 6 (1/r)| r=1 (p, q and s are non-negative integers, with superscripts denoting the power of the operator) are spherical harmonics of order 3p, 4q and 6s respectively. More generally, these operators may be combined as linear combination of T s 6 T q 4 T p 3 (1/r)| r=1 such that 6s + 4q + 3p = to obtain T-invariant spherical harmonic of order . The operator corresponding to this linear combination can be interpreted as a homogeneous polynomial of degree in T 3 , T 4 and T 6 , formally interpreted as an operator. Thus, for every homogeneous polynomial P ∈ H (where H denotes the module of homogenous polynomial of degree in T 3 , T 4 and T 6 ) a polynomial operator P → P can be defined via (28). We say P is homogeneous polynomial operator when P is a homogeneous polynomial. It follows from the discussion above that the function P(1/r)| r=1 is a T-invariant spherical harmonic. We now discuss the converse result [10].
Let us first note that if P 1 and P 2 are operators constructed from homogeneous polynomials P 1 , P 2 ∈ H , respectively such that P 1 = P 2 , then it does not necessarily follow that P 1 (1/r)| r=1 = P 2 (1/r)| r=1 . This is because some polynomial operators are trivially zero, since (x 2 +ŷ 2 +ẑ 2 )(1/r) = ∆(1/r) ≡ 0. However, if P 1 ≡ P 2 ( mod x 2 + y 2 + z 2 ), then indeed P 1 (1/r)| r=1 = P 2 (1/r)| r=1 . Therefore, only homogeneous polynomials modulo (x 2 + y 2 + z 2 ) in H lead to distinct spherical harmonics. For polynomial operators (30)-(32), it can be verified (computer algebra may be used for this purpose) that Thus, the powers of operator T 6 beyond 1 can be rewritten in terms of T 3 and T 4 . Based on this observation it can be shown (cf. [10] for details) that forms a basis for T-invariant spherical harmonic of order .

2.3.
Expressing in terms of standard spherical harmonics. We now discuss how spherical harmonics written using directional derivatives, discussed above, can be expressed in terms of the standard spherical harmonics, cf. (7). Let us define new operators,ξ where i 2 = −1. These are motivated by the following identities that relate directional derivatives to standard spherical harmonics [6]:

SANJAY DHARMAVARAM AND TIMOTHY J. HEALEY
Using (35) and the identity (x 2 +ŷ 2 +ẑ 2 ) ≡ 0, we obtain the relation We now show that every T-invariant spherical harmonic, previously expressed in (34) as a linear combination of directional derivatives, can be expressed as a linear combination of standard spherical harmonics. First note that since T 3 , T 4 and T 6 are each homogeous polymomial operators inx,ŷ andẑ, any operator that generates a T-invariant spherical harmonic of order , as in (34), can be written as where a mnp ∈ Z and m, n, p ∈ N such that m + n + p = . Substitutingx = (ξ +η)/2,ŷ = i(ξ −η)/2, we obtain Now observe that every term in (ξ +η) m (ξ −η) n can be expressed asξ jηk , such that j + k = m + n. Also, if α jk ∈ Z is the coefficient of termξ jηk , then it is clear that the coefficient ofξ kηj is (−1) n α jk . We can then rewrite (39) as where the summation involving j and k are such that j, k ∈ N, j + k = m + n. Define j = min{j, k}. By factoring (ξη) j from the term in brackets of (40) and using (37), we obtain P = m,n,p,j,k a m,n,p,j,k (i) n (−1) j ẑ p+2j ξ m+n−2j + (−1) nηm+n−2j , where a m,n,p,j,k := a m,n,p α jk /2 m+n . Note that every term in (41) has the form z u [ξ v ±η v ] for some u, v ∈ N. Thus, it follows from (36) that any T-invariant spherical harmonic constructed using operator P can be written in terms of standard spherical harmonics (7). We now apply this to operators T 3 , T 4 and T 6 , (cf. (30)-(32) ). Straightforward calculations using (35) and (37) yield Indeed, using (36) we verify (cf. (25)) that Similarly, applying (36) to (43) and (44), we obtain Here, I N = I × · · · × I (N times).
Z − 2n (odd n), Z n ⊕ Z c 2 (even n)ẑ 2p C qn (1/r)| r=1 ,ẑ 2p S qn (1/r)| r=1 2p + qn D d 2n (even n), D n ⊕ Z c 2 (odd n)ẑ 2p C 2qn (1/r)| r=1 ,ẑ 2p+1 S (2q+1)n (1/r)| r=1 2p + 2qn, 2p + 1 + (2q + 1)n (resp.) D d 2n (odd n), D n ⊕ Z c 2 (even n)ẑ 2p C qn (1/r)| r=1 2p + qn Table 1 shows the finite subgroups of O(3) (employing the notation of [3]) along with their respective functions that span the basis of the corresponding invariant spherical harmonic subspace, where p, q, j ∈ N ∪ {0} and s ∈ {0, 1}. The order of the spherical harmonic basis function shown in the second column is given in the last column. Note that some subgroups (rows 8, 9, 11, 12, 13 of Table 1) have invariant basis functions generated by both C and S operators, as seen in the second column. For these cases, the two entries on the third column represent the order of spherical harmonic functions for the two respective cases. Also, some subgroups (last four rows of Table 1) have the same basis functions, seen in the second column, depending on if n is even or odd. The operators that appear in the table are given by C n := (ξ n +η n ) (54) S n := i(ξ n −η n ) (55) These operators were obtained by expressing the ones considered in [10] in terms of (35). It is clear from (36) and the discussion in Section 2.3 that the basis functions of Table 1 can be expanded in terms of standard surface spherical harmonics.
Observe that for a given ∈ N, the dimension of the invariant subspace of spherical harmonics (with respect to a given group) is determined by the number of solutions to linear Diophantine equation obtained by setting the third column equal to . 4. Examples. We now present several examples, mostly illustrating the utility of the method for constructing bifurcation directions. In each case we use (36) to translate the operator representation for invariant spherical harmonics that appear in Table 1 to their corresponding standard spherical harmonic representation. We pick one or two examples from each of the discrete subgroups listed in Theorem 9.9 of [3]. In each case, we give the determining equation from Table 1, the explicit bifurcation direction (12), and a figure illustrating its nodal character. Table 1, row 5: 6p + 4q = 4 ⇒ p = 0, q = 1. Invariant bifurcation direction: cf. Figure 3(a).
whereũ( ) ∈ X G , for all sufficiently small , and where Y * is the bifurcation direction, cf. (12). Our complementary results presented herewith provide an efficient and systematic approach to the precise determination of Y * . We also mention that in problems like [4], [5], such local symmetric branches (57) are only the start of global bifurcating solution branches inheriting the same symmetry. Finally, we note the existence of local, O(3)-symmetry-breaking solutions exhibiting sub-maximal isotropy, cf. [2], [9]. These arise in the context of isotropy subgroups G ⊂ O(3) for which the invariant bifurcation problem on R × X G is not 1-dimensional. In contrast to the applications presented here based on the equivariant branching lemma, the existence of such solutions depends on a detailed analysis of the bifurcation equations. Although we do not carry out such analyses here, Examples 11 and 12 illustrate the construction of basis functions in the context of 2-dimensional local bifurcation equations. Finally we note that invariant spherical-harmonic bases for the planar subgroups of O(3), as well as the dimension of fixed-point subspaces for all subgroups and representations are provided in [2].