Orbit span of a design and some saturation theorems in commutative Schurian association schemes
Abstract
Orbit-span and dimension problems for designs have been studied in several classical association schemes using a variety of methods. Recently, through a detailed analysis of total trades, Ghorbani et al. showed that the orbit of a fixed combinatorial design asymptotically attains the full dimension permitted by the design equations. Analogous dimension-saturation results for the global spans of index-one designs in the bilinear forms and Grassmann schemes were obtained via laborious eigenvalue computations. In this paper, we work with the top fiber X of a graded poset carrying a compatible transitive action of a finite group G, and assume that the induced Schurian association scheme on X is commutative. For the multiplicity-free decomposition ℂX=U0⊕ U1⊕⋯⊕ Um, we prove that, under explicit spectral and quantitative conditions, the G-orbit of the characteristic vector of any t-design spans the maximal submodule U0⊕ Ut+1⊕ Ut+2⊕⋯⊕ Um allowed by the t-design definition. We verify these conditions asymptotically for the Hamming, bilinear forms, Johnson, and Grassmann schemes. This recovers the fixed-orbit saturation theorem of Ghorbani et al. for combinatorial designs, gives a new result for orthogonal arrays, and asymptotically extends the previous bilinear forms and Grassmann results to orbit spans of individual designs of arbitrary fixed index.
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BibTeX
@misc{li2026orbit,
title = {{Orbit span of a design and some saturation theorems in commutative Schurian association schemes}},
author = {Qilong Li and Yue Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13965},
eprint = {2609.13965},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13965},
}
RIS
TY - UNPB
TI - Orbit span of a design and some saturation theorems in commutative Schurian association schemes
AU - Qilong Li
AU - Yue Zhou
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13965
UR - https://arxiv.org/abs/2609.13965
AB - Orbit-span and dimension problems for designs have been studied in several classical association schemes using a variety of methods. Recently, through a detailed analysis of total trades, Ghorbani et al. showed that the orbit of a fixed combinatorial design asymptotically attains the full dimension permitted by the design equations. Analogous dimension-saturation results for the global spans of index-one designs in the bilinear forms and Grassmann schemes were obtained via laborious eigenvalue computations. In this paper, we work with the top fiber X of a graded poset carrying a compatible transitive action of a finite group G, and assume that the induced Schurian association scheme on X is commutative. For the multiplicity-free decomposition ℂ^X=U_0⊕ U_1⊕⋯⊕ U_m, we prove that, under explicit spectral and quantitative conditions, the G-orbit of the characteristic vector of any t-design spans the maximal submodule U_0⊕ U_t+1⊕ U_t+2⊕⋯⊕ U_m allowed by the t-design definition. We verify these conditions asymptotically for the Hamming, bilinear forms, Johnson, and Grassmann schemes. This recovers the fixed-orbit saturation theorem of Ghorbani et al. for combinatorial designs, gives a new result for orthogonal arrays, and asymptotically extends the previous bilinear forms and Grassmann results to orbit spans of individual designs of arbitrary fixed index.
ER -