The Υ_v,α,L,M(γ,Ω) are functions on the four-sphere S₄, which is parameterised by an angle γ in the range 0≤γ<π/3, and an SO(3) element Ω (which itself may be expressed in terms of three Euler angles). The Υ_v,α,L,M(γ,Ω) are orthonormal functions on S₄. They are expressed in terms of non-orthogonal functions Φ_N,t,L,M(γ,Ω). Each such function may itself expressed in the form

ΦN,t,L,M(γ,Ω) =∑ k≥0 Fk(NtL)(γ) ξ2k,M(L)(Ω),

(1)

where, in terms of Wigner D-functions, ξ_K,M^(L)(Ω)= (D_K,M^(L)(Ω) +(-1)^LD_-K,M^(L)(Ω))/ √(2(1+δ_K,0)). In this expression, F_k^(NtL)(γ) is independent of M. The data in this file presents expressions in this way for each spherical harmonic Υ_v,α,L,M(γ,Ω) with L≤20 and v≤45 (the much larger file presents expressions for L≤86 and v≤43).

The first line of the file gives the maximal values of L and v considered. After that, each value of L is considered in turn, and values f_i,j,k are defined, through which the F_k^(NtL)(γ) are obtained by

Fk(NtL)(γ) = ∑ 0≤i≤n1 ∑ 0≤j≤n2 fi,j,k (cosγ)n1-i (sinγ)i (cos2γ)n2-j (sin2γ)j (cos3γ)n3 (sin3γ)n4,

(2)

where n₁,n₂,n₃,n₄ are the unique solution to L=2n₁+2n₂+3n₄, N=n₁+2n₂+3n₃+3n₄, t=n₃. The f_i,j,k are listed for each n₁.n₂ with n₁+n₂=L/2 if L is even, and n₁+n₂=(L-3)/2 if L is odd. Each line defining these "header states" begins with (i,j), and is followed by the list of values [f_i,j,0,f_i,j,2,f_i,j,4,...]. Following this is listed, the set of all n₁.n₂.n₃.n₄ that correspond to the (N,L,t) with N of even parity, and N≤v_max. The corresponding functions Φ_N,t,L,M(γ,Ω) are then transformed into an orthonormal basis using the values listed in the triangular array next. If the Φ_i are the original functions in listed order, and the values in the i^th row of the triangular array are c_i,j then

Υi=κi ∑ 1≤j≤i ci,j Φj

(3)

is an SO(5) spherical harmonic, where κ_i is a normalisation factor, required so that Υ_i has unit norm. If the i^th value in the following line is η_i, this normalisation is given by κ_i=(1/π)√((2L+1)/8η_i). It remains to specify the indices v,α,L,M for which this Υ_i=Υ_v,α,L,M. The L index is that being considered, the M index can be chosen, and appears only as the second subscript of ξ_K,M^(L). The indices v and α are listed in parentheses alongside the n₁.n₂.n₃.n₄ designation for Φ_i listed previously. In fact, v=N for this function.

Having specified expressions Υ_v,α,L,M(γ,Ω) for even v (and N), expressions for odd v (and N) are specified in the same way for the same L. After that, subsequent values of L are dealt with.

The expressions described above were obtained using the algorithm developed in [RTR2004]. This was refined in [CRW2009]. It is also described in Chapter 4 of [RW2010]. They have been used, in particular, to calculate reduced SO(5)⊃SO(3) Clebsch-Gordan coefficients.

Last updated: 13/8/2014.