Quiver combinatorics for higher-dimensional triangulations
Quiver combinatorics for higher-dimensional triangulations
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高维三角剖分的 Quiver 组合学
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发表时间：
2021
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通讯作者：
Nicholas J. Williams
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作者：
Nicholas J. Williams
我们研究了由偶数维循环多面体的三角剖分产生的箭图的组合学。奥普曼和托马斯的工作将这种颤动精确地定位为高维星系团理论的原型。我们首先证明了一个$2d$维三角剖分没有内部$(d+1)$-单当且仅当它的箭图是在Iyama和Opmann意义下的$A$类型的割箭图。这是没有内部三角形的多边形的三角剖分对应于$A_{n}$dykin图的方向这一事实的高维推广。第一个结果的一个应用是，没有内部$(d+1)$单形的$2d$维循环多面体的三角剖分集通过双星翻转连接在一起--双星翻转是在四边形内翻转对角线的高维模拟。在大于2的维度中，无法在三角剖分中的所有位置执行双星翻转。我们的第二个结果给出了一个箭图理论准则，用于在$2D$维循环多面体的三角剖分上执行双星翻转。这为研究高维三角剖分的可变性提供了一个可视化工具，并指向了高维抖动突变理论可能是什么样子。事实上，我们应用这一结果给出了一个规则，用于在不一定是汇或源的顶点上突变切割颤动。
We investigate the combinatorics of quivers that arise from triangulations of even-dimensional cyclic polytopes. Work of Oppermann and Thomas pinpoints such quivers as the prototypes for higher-dimensional cluster theory. We first show that a $2d$-dimensional triangulation has no interior $(d + 1)$-simplices if and only if its quiver is a cut quiver of type $A$, in the sense of Iyama and Oppermann. This is a higher-dimensional generalisation of the fact that triangulations of polygons with no interior triangles correspond to orientations of an $A_{n}$ Dynkin diagram. An application of this first result is that the set of triangulations of a $2d$-dimensional cyclic polytope with no interior $(d + 1)$-simplices is connected via bistellar flips -- the higher-dimensional analogue of flipping a diagonal inside a quadrilateral. In dimensions higher than 2, bistellar flips cannot be performed at all locations in a triangulation. Our second result gives a quiver-theoretic criterion for performing bistellar flips on a triangulation of a $2d$-dimensional cyclic polytope. This provides a visual tool for studying mutability of higher-dimensional triangulations and points towards what a theory of higher-dimensional quiver mutation could look like. Indeed, we apply this result to give a rule for mutating cut quivers at vertices which are not necessarily sinks or sources.