For a finite subset $X$ of the $d$-dimensional unit sphere, the harmonic strength $T(X)$ of $X$ is the set of $\ell \in \mathbb{N}$ such that $\sum _{x\in X} P(x)=0$ for all harmonic polynomials $P$ of homogeneous degree $\ell $. We will study three exceptional finite groups of unit quaternions, called the binary tetrahedral group $2T$ of order 24, the octahedral group $2O$ of order 48, and the icosahedral group $2I$ 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 $2O$ is unique as a minimal subset $X$ of the 3-dimensional unit sphere with $T(X)=\lbrace 22,14,10,6,4,2 \rbrace \cup \mathbb{O}^+$, where $\mathbb{O}^+$ denotes the set of all positive odd integers. This result provides the first characterization of $2O$ from the spherical design viewpoint.
For $G\in \lbrace 2T,2O,2I\rbrace $, we consider the lattice $\mathcal{O}_{G}$ generated by $G$ over $R_G$ on which the group $G$ acts by multiplication, where $R_{2T}=\mathbb{Z},\ R_{2O}=\mathbb{Z}[\sqrt{2}],\ R_{2I}=\mathbb{Z}[(1+\sqrt{5})/2]$ are the rings of integers. We introduce the spherical theta function $\theta _{G,P}(z)$ attached to the lattice $\mathcal{O}_G$ and a harmonic polynomial $P$ of degree $\ell $ and prove that they are modular forms. By applying our results on the characterization of $G$ as a spherical design, we determine the cases in which the $\mathbb{C}$-vector space spanned by all $\theta _{G,P}(z)$ of harmonic polynomials $P$ of homogeneous degree $\ell $ has dimension zero–without relying on the theory of modular forms.
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Keywords: Finite quaternionic unit group, spherical designs of harmonic index, harmonic strength, Linear programming bound, uniqueness, spherical theta functions, modular forms
Hirao, Masatake 1 ; Nozaki, Hiroshi 2 ; Tasaka, Koji 3
CC-BY 4.0
Hirao, Masatake; Nozaki, Hiroshi; Tasaka, Koji. Spherical designs for finite quaternionic unit groups and their applications to modular forms. Algebraic Combinatorics, Volume 9 (2026) no. 4, pp. 1157-1178. doi: 10.5802/alco.505
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author = {Hirao, Masatake and Nozaki, Hiroshi and Tasaka, Koji},
title = {Spherical designs for finite quaternionic unit groups and their applications to modular forms},
journal = {Algebraic Combinatorics},
pages = {1157--1178},
year = {2026},
publisher = {The Combinatorics Consortium},
volume = {9},
number = {4},
doi = {10.5802/alco.505},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.505/}
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TY - JOUR AU - Hirao, Masatake AU - Nozaki, Hiroshi AU - Tasaka, Koji TI - Spherical designs for finite quaternionic unit groups and their applications to modular forms JO - Algebraic Combinatorics PY - 2026 SP - 1157 EP - 1178 VL - 9 IS - 4 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.505/ DO - 10.5802/alco.505 LA - en ID - ALCO_2026__9_4_1157_0 ER -
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