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 $\mathbb{C}^{X}=U_{0}\oplus U_{1}\oplus\cdots\oplus U_{m}$, we prove that, under explicit spectral and quantitative conditions, the $G$-orbit of the characteristic vector of any $t$-design spans the maximal submodule $U_{0}\oplus U_{t+1}\oplus U_{t+2}\oplus\cdots\oplus U_{m}$ 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.