关联方案与极空间各向异性点的正交图
Association schemes and orthogonality graphs on anisotropic points of polar spaces
AI总结：
本文研究了极空间各向异性点上的关联方案与正交图的谱，通过定义基于点间直线类型和二次类型的关联关系，得到非原始5类关联方案，并计算了相关特征矩阵。
摘要：
在本文中，我们研究经典极空间各向异性点上的关联方案。我们的主要结果涉及PG(n,q)中非退化椭圆和双曲二次曲面，其中q为奇数。我们定义了这些二次曲面各向异性点上的关系，这些关系取决于由点所张成的直线类型以及这些点是否具有相同的“二次类型”。这导致了一个非原始的5类关联方案。我们计算了该方案的特征矩阵和对偶特征矩阵。我们还利用这一结果，结合文献中关于其他经典极空间的类似结果，精确计算了非退化二次曲面在奇特征下各向异性点的正交图的谱，以及非退化赫米特变种的谱。作为副产品，我们得到了非退化赫米特变种各向异性点上的3类关联方案，其中包含两个点的关系取决于由这些点张成的直线类型以及它们是否正交。
英文摘要：
In this paper, we study association schemes on the anisotropic points of classical polar spaces. Our main result concerns non-degenerate elliptic and hyperbolic quadrics in PG$(n,q)$ with $q$ odd. We define relations on the anisotropic points of such a quadric that depend on the type of line spanned by the points and whether or not they are of the same "quadratic type". This yields an imprimitive $5$-class association scheme. We calculate the matrices of eigenvalues and dual eigenvalues of this scheme. We also use this result, together with similar results from the literature concerning other classical polar spaces, to exactly calculate the spectrum of orthogonality graphs on the anisotropic points of non-degenerate quadrics in odd characteristic and of non-degenerate Hermitian varieties. As a byproduct, we obtain a $3$-class association scheme on the anisotropic points of non-degenerate Hermitian varieties, where the relation containing two points depends on the type of line spanned by these points, and whether or not they are orthogonal.
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