Flow polytopes of signed graphs and the Kostant partition function
Flow polytopes of signed graphs and the Kostant partition function
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有符号图的流多胞形和 Kostant 配分函数
DOI：
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发表时间：
2012
期刊：
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通讯作者：
A. Morales
中科院分区：
文献类型：
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作者：
Karola Mészáros;A. Morales
我们建立了与符号图和Kostant配分函数相关联的流多面体的体积之间的关系。巴尔多尼和韦尔涅使用留数技术详细研究了这种关系的一种特殊情况，即当图无符号时。与他们的方法相比，我们提供了完全组合的证据，灵感来自Postnikov和斯坦利的工作流多面体。作为一类特殊的流多面体，我们研究了Chan-Robbins-Yuen多面体。受$A_n$型的体积公式$\prod_{k=1}^{n-2} Cat（k）$（其中$Cat（k）$是第k个Catalan数）的启发，我们引入了$C_{n+1}$和$D_{n+1}$ Chan-Robbins-Yuen型多胞形沿着有趣的性质.
We establish the relationship between volumes of flow polytopes associated to signed graphs and the Kostant partition function. A special case of this relationship, namely, when the graphs are signless, has been studied in detail by Baldoni and Vergne using techniques of residues. In contrast with their approach, we provide entirely combinatorial proofs inspired by the work of Postnikov and Stanley on flow polytopes. As a fascinating special family of flow polytopes, we study the Chan-Robbins-Yuen polytopes. Motivated by the beautiful volume formula $\prod_{k=1}^{n-2} Cat(k)$ for the type $A_n$ version, where $Cat(k)$ is the $k$th Catalan number, we introduce type $C_{n+1}$ and $D_{n+1}$ Chan-Robbins-Yuen polytopes along with intriguing conjectures pertaining to their properties.