Zonotopal代数
Zonotopal algebra
摘要：
一个给定的线性自同态$X$的$\R^N$的几何和组合性质的财富被捕获在其相关的zonotope $Z(X)$的研究，并通过对偶，其相关的超平面安排${\cal H}(X)$。这一众所周知的研究路线在病例$n\eqbd\rank X \ll N$中特别有趣。我们将这一研究提高到代数水平，并将$X$与三种代数结构相关联，在本文中称为{ \it外部、中心和内部}。每一个代数结构给出的一对齐次多项式理想的$n$变量是对偶的：一个编码属性的安排${\cal H}(X)$，而另一个编码的对偶性质的zonotope $Z(X)$。的代数结构定义纯粹的组合结构的$X$，但随后被证明是同样可获得的通过应用适当的代数分析操作的$Z(X)$或${\cal H}(X)$。这个理论是普遍的，因为它不需要对映射$X$做任何假设（唯一的例外是，对$Z(X)$的代数-解析运算只在$X$是幺模的情况下才能得到所寻求的结果），并且提供了新的工具，可以用于枚举组合学、图论、表示论、多面体几何和逼近论。
英文摘要：
A wealth of geometric and combinatorial properties of a given linear endomorphism $X$ of $\R^N$ is captured in the study of its associated zonotope $Z(X)$, and, by duality, its associated hyperplane arrangement ${\cal H}(X)$. This well-known line of study is particularly interesting in case $n\eqbd\rank X \ll N$. We enhance this study to an algebraic level, and associate $X$ with three algebraic structures, referred herein as {\it external, central, and internal.} Each algebraic structure is given in terms of a pair of homogeneous polynomial ideals in $n$ variables that are dual to each other: one encodes properties of the arrangement ${\cal H}(X)$, while the other encodes by duality properties of the zonotope $Z(X)$. The algebraic structures are defined purely in terms of the combinatorial structure of $X$, but are subsequently proved to be equally obtainable by applying suitable algebro-analytic operations to either of $Z(X)$ or ${\cal H}(X)$. The theory is universal in the sense that it requires no assumptions on the map $X$ (the only exception being that the algebro-analytic operations on $Z(X)$ yield sought-for results only in case $X$ is unimodular), and provides new tools that can be used in enumerative combinatorics, graph theory, representation theory, polytope geometry, and approximation theory.
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