Tight relative t-designs on two shells in hypercubes, and Hahn and Hermite polynomials
Tight relative t-designs on two shells in hypercubes, and Hahn and Hermite polynomials
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DOI：
10.26493/1855-3974.2352.eaf
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发表时间：
2020-06
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影响因子：
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通讯作者：
E. Bannai;E. Bannai;Hajime Tanaka;Yan Zhu
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文献类型：
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作者：
E. Bannai;E. Bannai;Hajime Tanaka;Yan Zhu
n维超立方体$\mathcal{Q}_n$中的相对$t$-设计等价于加权正则$t$-平衡设计，它通过允许多个块大小和权重来推广组合$t$-$（n，k，\lambda）$设计。部分受最近关于同心球上紧欧氏$t$-设计的研究的启发，本文讨论了两壳上$\mathcal{Q}_n$中的紧相对$t$-设计.我们表明，在一个温和的条件下，这样一个相对的$t$-设计诱导结构的相干配置与两个纤维。此外，从这个结构中，我们推出的多项式族的哈恩超几何正交多项式必须只有整单零点。Terwilliger代数是建立这些结果的主要工具。通过对Hahn多项式在适当的极限过程下退化为Hermite多项式时零点的性质的研究，我们证明了一个定理，该定理给出了支持在两个壳上的$\mathcal{Q}_n$中的非平凡紧相对$t$-设计对于大$t$是罕见的部分证据.
Relative $t$-designs in the $n$-dimensional hypercube $\mathcal{Q}_n$ are equivalent to weighted regular $t$-wise balanced designs, which generalize combinatorial $t$-$(n,k,\lambda)$ designs by allowing multiple block sizes as well as weights. Partly motivated by the recent study on tight Euclidean $t$-designs on two concentric spheres, in this paper we discuss tight relative $t$-designs in $\mathcal{Q}_n$ supported on two shells. We show under a mild condition that such a relative $t$-design induces the structure of a coherent configuration with two fibers. Moreover, from this structure we deduce that a polynomial from the family of the Hahn hypergeometric orthogonal polynomials must have only integral simple zeros. The Terwilliger algebra is the main tool to establish these results. By explicitly evaluating the behavior of the zeros of the Hahn polynomials when they degenerate to the Hermite polynomials under an appropriate limit process, we prove a theorem which gives a partial evidence that the non-trivial tight relative $t$-designs in $\mathcal{Q}_n$ supported on two shells are rare for large $t$.