Spherical designs for finite quaternionic unit groups and their applications to modular forms

For a finite subset X of the d -dimensional unit sphere, the harmonic strength T ( X ) of X is the set of ℓ ∈ ℕ such that ∑ x ∈ X P ( x ) = 0 for all harmonic polynomials P of homogeneous degree ℓ . We will study three exceptional finite groups of unit quaternions, called the binary tetrahedral group 2 T of order 24, the octahedral group 2 O of order 48, and the icosahedral group 2 I of order 120, which can be viewed as a subset of the 3-dimensional unit sphere. For these three groups, we determine the harmonic strength and show the minimality and the uniqueness as spherical designs. In particular, the group 2 O is unique as a minimal subset X of the 3-dimensional unit sphere with T ( X ) = { 22 , 14 , 10 , 6 , 4 , 2 } ∪ 𝕆 + , where 𝕆 + denotes the set of all positive odd integers. This result provides the first characterization of 2 O from the spherical design viewpoint. For G ∈ { 2 T , 2 O , 2 I } , we consider the lattice 𝒪 G generated by G over R G on which the group G acts by multiplication, where R 2 T = ℤ , R 2 O = ℤ [ 2 ] , R 2 I = ℤ [