arXiv 2112.091892021-12-16math.COmath.RT

高维三角剖分的箭图组合

Quiver combinatorics for higher-dimensional triangulations

Nicholas J. Williams

备注：: 26 pages, 9 figures

摘要：

我们研究了偶维循环多面体三角剖分产生的箭图的组合学。Oppermann和Thomas的工作精确地指出了这种颤动是高维簇理论的原型。我们首先证明了一个$2d$维三角剖分没有内部$(d + 1)$-单形当且仅当它的单形是一个$A$型的截单形，在Iyama和Oppermann的意义下.这是一个更高维的推广的事实，三角形的多边形没有内部三角形对应的方向的$A_{n}$ Dynkin图。这第一个结果的应用是，没有内部$(d + 1)$-单纯形的$2d$维循环多面体的三角剖分集通过bistellar翻转连接-翻转四边形内部的对角线的高维模拟。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颤抖理论的标准进行bistellar翻转的三角剖分的$2d$维循环多面体。这为研究高维三角剖分的可变性提供了一个可视化的工具，并指出了高维三角剖分的突变理论可能是什么样子。事实上，我们应用这一结果给一个规则突变切颤动的顶点不一定是汇或源。

英文摘要：

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.

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