The Zonotopal Algebra of the Broken Wheel Graph and its Generalization

The machinery of zonotopal algebra is linked with two particular polytopes: the Stanley-Pitman polytope and the regular simplex ${\mathfrak{S}\mathfrak{i}\mathfrak{m}}_n(t_1,\mathrm{.}\mathrm{.}\mathrm{.},t_n)$ with parameters $t_1,…,t_n\in $\mathbb{R}$+^n$,definedbytheinequalities$\sum{i=1}^n r_i\leq \sum_{i=1}^n t_i, \mbox{ } r_i\in $\mathbb{R}$+^n,$wherethe$(r_i){i\in [n]}$ are variables. Specifically, we will discuss the central Dahmen-Micchelli space of the broken wheel graph $B{W}_n$ and its dual, the $\mathcal{P}$-central space. We will observe that the $\mathcal{P}$-central space of $B{W}_n$ is monomial, with a basis given by the $B{W}_n$-parking functions. We will show that the volume polynomial of the the Stanley-Pitman polytope lies in the central Dahmen-Micchelli space of $B{W}_n$ and is precisely the polynomial in a particular basis of the central Dahmen-Micchelli space which corresponds to the monomial $t_1{t}_2\cdots t_n$ in the dual monomial basis of the $\mathcal{P}$-central space. We will then define the generalized broken wheel graph $GB{W}_n(T)$ 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 $\mathit{\text{generalized}}\mathit{\text{broken}}\mathit{\text{wheel}}\mathit{\text{graphs}}$. Each generalized broken wheel graph constructed from $T$ will give us a polytope, its volume polynomial, and a $\mathit{\text{reference}}\mathit{\text{monomial}}$. The $2^{n-1}$ polytopes together give a polyhedral subdivision of ${\mathfrak{S}\mathfrak{i}\mathfrak{m}}_n(t_1,\mathrm{.}\mathrm{.}\mathrm{.},t_n)$, 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.