AI 中文总结
本文研究自由多面体的次齐性与Arveson边界，完成三维锥的次齐性分类，推导其传递性，确定两类单纯形乘积上自由多面体的边界点数量，给出应用分类并构造兼具两类行为的锥。
AI 中文摘要
我们研究多面锥上极小算子系统的次齐性，等价于自由多面体的不可约Arveson边界点的数量。我们在三维锥（对应多面体的二维）中得到完整分类：具有三条极射线的锥是1-次齐性的，具有四条极射线的锥是2-次齐性的，而具有至少五条极射线的锥不是次齐性的。在后一情形中，我们在每个偶数矩阵层级上构造了不可约Arveson边界点。我们证明次齐性可传递到面和面商，还确定了两个单纯形乘积上自由多面体的不可约Arveson边界点数量，得到了两段乘积与其余所有情形之间的二分性。作为应用，我们对具有n+1条极射线的n维锥进行分类；更一般地，当每个面最多遗漏两条极射线时，我们证明次齐性恰好出现在单纯锥与三维四射线锥的直和中。我们还在一系列固定维度和射线数量下，构造了同时表现出次齐性和非次齐性行为的锥。
英文摘要
We study subhomogeneity of the minimal operator system over a polyhedral cone, or equivalently the size of irreducible Arveson boundary points of free polyhedra. We obtain a complete classification in dimension three for cones (equivalently, dimension two for polytopes): cones with three extreme rays are 1-subhomogeneous, cones with four extreme rays are 2-subhomogeneous, and cones with at least five extreme rays are not subhomogeneous. In the last case, we construct irreducible Arveson boundary points at every even matrix level. We prove that subhomogeneity passes to faces and face quotients. We also determine the size of irreducible Arveson boundary points of free polyhedra over products of two simplices, obtaining a dichotomy between the product of two segments and all remaining cases. As applications, we classify n-dimensional cones with n+1 extreme rays. More generally, when every facet omits at most two extreme rays, we show that subhomogeneity occurs exactly for direct sums of simplicial cones and three-dimensional four-ray cones. We also construct, for a range of fixed dimensions and ray counts, cones exhibiting both subhomogeneous and non-subhomogeneous behavior.
