Quadratic-Defect Completions of Spherical $2$-Design Orbits
Abstract
We study how spherical $2$-designs arising from finite group orbits can be completed to spherical $4$-designs by adjoining further orbits, allowing weights in the general theory. For an irreducible real orthogonal G-module W with π»=EndG(W)β{β,β,β}, we consider the multiplicity-two representation Wβ W and retain the failure of the $2$-design equation M*M=1/2 I2 as a quadratic defect. When the invariant quartics are determined by the Hermitian Gram matrix, the fourth-moment problem reduces to a mean and covariance condition on these defects. This yields a sharp lower bound for the total weight of the correction orbits; at equality, their normalized defects form a weighted spherical $2$-design in the associated defect space. The quartic condition holds for the multiqubit Clifford groups in every dimension rβ₯ 1, giving an unbounded-dimensional family with a fixed three-dimensional defect space; among equality cases using the minimum number of correction orbits, the defect geometry is always a regular tetrahedron. As a complementary unweighted example, we construct a $378$-point W(E6)-invariant spherical $4$-design in S11 and prove that it is sharp among unweighted invariant completions containing a spherical $2$-design orbit.
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BibTeX
@misc{chien2026quadraticdefect,
title = {{Quadratic-Defect Completions of Spherical $2$-Design Orbits}},
author = {Kuan-Cheng Chien and Ming-Hsuan Kang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25032},
eprint = {2609.25032},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25032},
}
RIS
TY - UNPB
TI - Quadratic-Defect Completions of Spherical $2$-Design Orbits
AU - Kuan-Cheng Chien
AU - Ming-Hsuan Kang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.25032
UR - https://arxiv.org/abs/2609.25032
AB - We study how spherical $2$-designs arising from finite group orbits can be completed to spherical $4$-designs by adjoining further orbits, allowing weights in the general theory. For an irreducible real orthogonal G-module W with π»=End_G(W)β{β,β,β}, we consider the multiplicity-two representation Wβ W and retain the failure of the $2$-design equation M^*M=1/2 I_2 as a quadratic defect. When the invariant quartics are determined by the Hermitian Gram matrix, the fourth-moment problem reduces to a mean and covariance condition on these defects. This yields a sharp lower bound for the total weight of the correction orbits; at equality, their normalized defects form a weighted spherical $2$-design in the associated defect space. The quartic condition holds for the multiqubit Clifford groups in every dimension rβ₯ 1, giving an unbounded-dimensional family with a fixed three-dimensional defect space; among equality cases using the minimum number of correction orbits, the defect geometry is always a regular tetrahedron. As a complementary unweighted example, we construct a $378$-point W(E_6)-invariant spherical $4$-design in S^11 and prove that it is sharp among unweighted invariant completions containing a spherical $2$-design orbit.
ER -