设计的轨道跨度与交换Schurian结合方案中的若干饱和定理

中文摘要 AI 辅助

在若干经典结合方案中，设计（design）的轨道跨度问题与维数问题已通过多种方法得到研究。近期，Ghorbani等人通过对总交易（total trades）的细致分析，证明了固定组合设计的轨道渐近地达到设计方程所允许的满维数。对于双线性型方案（bilinear forms scheme）和Grassmann方案（Grassmann scheme）中指数为一的设计的整体跨度，类似的维数饱和结果是通过繁重的特征值计算获得的。在本文中，我们考虑一个带有有限群G的相容传递作用的分级偏序集的顶部纤维X，并假设X上诱导的Schurian结合方案是交换的。对于多重性自由分解C^X=U_0⊕U_1⊕…⊕U_m，我们证明，在明确的谱条件和定量条件下，任何t-设计的特征向量的G-轨道张成由t-设计定义所允许的最大子模U_0⊕U_{t+1}⊕U_{t+2}⊕…⊕U_m。我们渐近地验证了Hamming方案、双线性型方案、Johnson方案和Grassmann方案中的这些条件。这恢复了Ghorbani等人关于组合设计的固定轨道饱和定理，给出了正交阵列（orthogonal arrays）的新结果，并将先前的双线性型和Grassmann结果渐近地推广到任意固定指数的单个设计的轨道跨度。

英文摘要

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.

发表机构

• College of Science, National University of Defense Technology（国防科技大学理学院）

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