断轮图的Zonotopal代数及其推广
The Zonotopal Algebra of the Broken Wheel Graph and its Generalization
摘要：
带蛋白石代数的机制与两个特殊的多面体有关：斯坦利-皮特曼多面体和正则单形$\mathfrak{Sim}_n(t_1,...,t_n)$，参数为$t_1,...,t_n\in \mathbb{R}_+^n$，由不等式$\sum_{i=1}^n r_i\leq \sum_{i=1}^n t_i, \mbox{ } r_i\in \mathbb{R}_+^n,$定义，其中$(r_i)_{i\in [n]}$是变量。具体来说，我们将讨论断轮图$BW_n$的中心Dahmen-Micchelli空间及其对偶空间，即$\mathcal{P}$ -中心空间。我们将观察到$BW_n$的$\mathcal{P}$ -中心空间是单项式的，其基由$BW_n$ -parking函数给出。我们将证明Stanley-Pitman多面体的体积多项式位于$BW_n$的中心Dahmen-Micchelli空间中，并且恰好是中心Dahmen-Micchelli空间的特定基中的多项式，该特定基对应于$\mathcal{P}$ -中心空间的对偶单项基中的单项$t_1t_2\cdots t_n$。然后，我们将定义广义破轮图$GBW_n(T)$为一个给定的根树$T$上的$n$顶点。对于每个这样的树，我们可以构造$2^{n-1}$有向图，我们将其称为\textit{generalized broken wheel graphs}。从$T$构造的每个广义断轮图将给我们一个多面体，它的体积多项式，和一个\textit{reference monomial}。$2^{n-1}$多面体一起给出了$\mathfrak{Sim}_n(t_1,...,t_n)$的多面体细分，它们的体积多项式一起给出了相应的中心Dahmen-Micchelli空间的次数为$n$的齐次多项式的子空间的基，并且它们的参考单项式一起给出了其对偶的基。
英文摘要：
The machinery of zonotopal algebra is linked with two particular polytopes: the Stanley-Pitman polytope and the regular simplex $\mathfrak{Sim}_n(t_1,...,t_n)$ with parameters $t_1,...,t_n\in \mathbb{R}_+^n$, defined by the inequalities $\sum_{i=1}^n r_i\leq \sum_{i=1}^n t_i, \mbox{ } r_i\in \mathbb{R}_+^n,$ where the $(r_i)_{i\in [n]}$ are variables. Specifically, we will discuss the central Dahmen-Micchelli space of the broken wheel graph $BW_n$ and its dual, the $\mathcal{P}$-central space. We will observe that the $\mathcal{P}$-central space of $BW_n$ is monomial, with a basis given by the $BW_n$-parking functions. We will show that the volume polynomial of the the Stanley-Pitman polytope lies in the central Dahmen-Micchelli space of $BW_n$ and is precisely the polynomial in a particular basis of the central Dahmen-Micchelli space which corresponds to the monomial $t_1t_2\cdots t_n$ in the dual monomial basis of the $\mathcal{P}$-central space. We will then define the generalized broken wheel graph $GBW_n(T)$ for a given rooted tree $T$ on $n$ vertices. For every such tree, we can construct $2^{n-1}$ directed graphs, which we will refer to as \textit{generalized broken wheel graphs}. Each generalized broken wheel graph constructed from $T$ will give us a polytope, its volume polynomial, and a \textit{reference monomial}. The $2^{n-1}$ polytopes together give a polyhedral subdivision of $\mathfrak{Sim}_n(t_1,...,t_n)$, their volume polynomials together give a basis for the subspace of homogeneous polynomials of degree $n$ of the corresponding central Dahmen-Micchelli space, and their reference monomials together give a basis for its dual.
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