Hierarchical zonotopal spaces

TL;DR

This paper extends zonotopal algebra by developing a hierarchy of new combinatorially defined spaces, broadening the foundational principles beyond the traditional nested ideal pairs.

Contribution

It introduces a hierarchical framework for zonotopal spaces, expanding the algebraic and combinatorial structures beyond the classical setup.

Findings

Established a hierarchy of zonotopal spaces

Extended algebraic structures beyond traditional ideal pairs

Provided combinatorial methods for new space construction

Abstract

Zonotopal algebra interweaves algebraic, geometric and combinatorial properties of a given linear map X. Of basic significance in this theory is the fact that the algebraic structures are derived from the geometry (via a non-linear procedure known as "the least map"), and that the statistics of the algebraic structures (e.g., the Hilbert series of various polynomial ideals) are combinatorial, i.e., computable using a simple discrete algorithm known as "the valuation function". On the other hand, the theory is somewhat rigid since it deals, for the given X, with exactly two pairs each of which is made of a nested sequence of three ideals: an external ideal (the smallest), a central ideal (the middle), and an internal ideal (the largest). In this paper we show that the fundamental principles of zonotopal algebra as described in the previous paragraph extend far beyond the setup of external, central and internal ideals by building a whole hierarchy of new combinatorially defined zonotopal spaces.