Quiver combinatorics and triangulations of cyclic polytopes
Quiver combinatorics and triangulations of cyclic polytopes
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DOI：
10.5802/alco.280
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发表时间：
2023-06
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通讯作者：
Nicholas J. Williams
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文献类型：
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作者：
Nicholas J. Williams
受高等同调代数的启发，我们将箭图与偶数维循环多面体的三角剖分联系起来，并证明了两个结果，表明三角剖分的信息是如何被编码在箭图中的。我们首先证明了Iyama和Oppermann的割箭图精确地对应于没有内部（d + 1）-单形的二维三角剖分。这意味着这些三角剖分形成翻转图的连通子图。我们的第二个结果显示了如何三角剖分的三角剖分可以用来识别可变的内部d -单形。这指出了一个更高维度的突变理论可能看起来像什么，并给出了一个新的方式来理解翻转的三角剖分的偶数维循环多面体。
Motivated by higher homological algebra, we associate quivers to triangulations of even-dimensional cyclic polytopes and prove two results showing what information about the triangulation is encoded in the quiver. We first show that the cut quivers of Iyama and Oppermann correspond precisely to 2 d -dimensional triangulations without interior ( d + 1)- simplices. This implies that these triangulations form a connected subgraph of the flip graph. Our second result shows how the quiver of a triangulation can be used to identify mutable internal d -simplices. This points towards what a theory of higher-dimensional quiver mutation might look like and gives a new way of understanding flips of triangulations of even-dimensional cyclic polytopes.