h-多项式的约化形式
h-Polynomials via Reduced Forms
摘要：
流多面体$\mathcal{F}_{\widetilde{G}}$是图$\widetilde{G}$上的非负单位流的集合。流多面体的细分代数规定了一种将流多面体$\mathcal{F}_{\widetilde{G}}$分解成单形的方法。这样的分解是由单项$\prod_{(i,j)\in E(G)}x_{ij}$的所谓简化形式的项编码的。证明了利用流多面体的剖分代数不仅可以构造流多面体的剖分，而且可以构造流多面体的正则旗三角剖分。证明了细分代数中的约化形式是流多面体三角剖分的$h$ -多项式的推广。我们推导出上述结果的几个推论，最值得注意的是证明了Kirillov关于非交换准经典Yang-Baxter代数中约化形式的非负性的猜想的某些情况。
英文摘要：
The flow polytope $\mathcal{F}_{\widetilde{G}}$ is the set of nonnegative unit flows on the graph $\widetilde{G}$. The subdivision algebra of flow polytopes prescribes a way to dissect a flow polytope $\mathcal{F}_{\widetilde{G}}$ into simplices. Such a dissection is encoded by the terms of the so called reduced form of the monomial $\prod_{(i,j)\in E(G)}x_{ij}$. We prove that we can use the subdivision algebra of flow polytopes to construct not only dissections, but also regular flag triangulations of flow polytopes. We prove that reduced forms in the subdivision algebra are generalizations of $h$-polynomials of the triangulations of flow polytopes. We deduce several corollaries of the above results, most notably proving certain cases of a conjecture of Kirillov about the nonnegativity of reduced forms in the noncommutative quasi-classical Yang-Baxter algebra.
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