2018/10/10 by Brodsky, Sarah B.
#Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1810.04432
The machinery of zonotopal algebra is linked with two particular polytopes: the Stanley-Pitman polytope and the regular simplex \mathfrakSimn(t1,...,tn) with parameters t1,...,tn∈ ℝ+n, defined by the inequalities ∑i=1n ri≤ ∑i=1n ti, ri∈ ℝ+n, where the (ri)i∈ [n] are variables. Specifically, we will discuss the central Dahmen-Micchelli space of the broken wheel graph BWn and its dual, the P-central space. We will observe that the P-central space of BWn is monomial, with a basis given by the BWn-parking functions. We will show that the volume polynomial of the the Stanley-Pitman polytope lies in the central Dahmen-Micchelli space of BWn and is precisely the polynomial in a particular basis of the central Dahmen-Micchelli space which corresponds to the monomial t1t2⋯ tn in the dual monomial basis of the P-central space. We will then define the generalized broken wheel graph GBWn(T) for a given rooted tree T on n vertices. For every such tree, we can construct 2n-1 directed graphs, which we will refer to as generalized broken wheel graphs. Each generalized broken wheel graph constructed from T will give us a polytope, its volume polynomial, and a reference monomial. The 2n-1 polytopes together give a polyhedral subdivision of \mathfrakSimn(t1,...,tn), 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.