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PreprintCC BY 4.0arXiv

Dual near-factorizations and Schur products in association schemes

  • Bobby Miraftab
  • Sho Suda
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Abstract

Let G and H be Gram matrices of spherical $2$-designs on the same number of indices. We study when their Schur product is again the Gram matrix of a spherical design. We give a criterion in terms of orthogonal projections with constant diagonal, and an explicit $15$-point spherical $3$-design whose Gram matrix belongs to no symmetric Bose-Mesner algebra. For a symmetric association scheme, the factorization problem takes the form v(ES∘ ET)=μEU. This is a counterpart of ordinary matrix product factorizations in the adjacency basis. The case EU=I−E0 gives a dual near-factorization and has a regular simplex as its normalized output. We characterize complementary input supports by a strongly regular fusion, give a reciprocal criterion, and classify the case in which one factor is the last Hamming idempotent.

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Cite

BibTeXRIS
BibTeX
@misc{miraftab2026dual,
  title = {{Dual near-factorizations and Schur products in association schemes}},
  author = {Bobby Miraftab and Sho Suda},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34598},
  eprint = {2610.34598},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34598},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Dual near-factorizations and Schur products in association schemes
AU  - Bobby Miraftab
AU  - Sho Suda
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2610.34598
UR  - https://arxiv.org/abs/2610.34598
AB  - Let G and H be Gram matrices of spherical $2$-designs on the same number of indices. We study when their Schur product is again the Gram matrix of a spherical design. We give a criterion in terms of orthogonal projections with constant diagonal, and an explicit $15$-point spherical $3$-design whose Gram matrix belongs to no symmetric Bose-Mesner algebra. For a symmetric association scheme, the factorization problem takes the form v(E_S∘ E_T)=μE_U. This is a counterpart of ordinary matrix product factorizations in the adjacency basis. The case E_U=I−E_0 gives a dual near-factorization and has a regular simplex as its normalized output. We characterize complementary input supports by a strongly regular fusion, give a reciprocal criterion, and classify the case in which one factor is the last Hamming idempotent.
ER  -