The Zonotopal Algebra of the Broken Wheel Graph and its Generalization

TL;DR

This paper explores the connections between zonotopal algebra, the broken wheel graph, and certain polytopes, introducing a generalized framework that links graph structures, polytopes, and polynomial spaces.

Contribution

It introduces the generalized broken wheel graph for any rooted tree, linking graph theory, polytopes, and polynomial spaces in a novel unified framework.

Findings

The volume polynomial of the Stanley-Pitman polytope lies in the central Dahmen-Micchelli space.

The $ ext{P}$-central space of the broken wheel graph is monomial with a basis of parking functions.

Generalized broken wheel graphs produce a polyhedral subdivision with basis polynomials and dual monomials.

Abstract

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.