The Generalized External Order and Applications to Zonotopal Algebra Bryan R. Gillespie

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• Author: Bryan R. Gillespie

Extrapolating from the work of Las Vergnas on the external active order for matroid bases, and inspired by the structure of Lenz’s forward exchange matroids in the theory of zonotopal algebra, we develop the combinatorial theory of the generalized external order. This partial ordering on the independent sets of an ordered matroid is a supersolvable join-distributive lattice which is a refinement of the geometric lattice of flats, and is fundamentally derived from the classical notion of matroid activity. We uniquely classify the lattices which occur as the external order of an ordered matroid, and we explore the intricate structure of the lattice’s downward covering relations, as well as its behavior under deletion and contraction of the underlying matroid.

Table of contents:

1 Introduction

2.1 Matroids and Antimatroids

2.2 Zonotopal Algebra

3 The Generalized External Order

3.1 Feasible and Independent Sets of Join-distributive Lattices

3.2 Definition and Fundamental Properties

3.3 Lattice Theoretic Classification

3.4 Deletion and Contraction

3.5 Passive Exchanges and Downward Covering Relations

4 Applications to Zonotopal Algebra

4.1 Forward Exchange Matroids and the External Order

4.2 Canonical Basis Polynomials

4.3 Differential Structure of CentralD-Polynomials

4.4 Recursive Construction for the CentralD-Basis

4.5 Explicit Construction for the Internal and Semi-internalP-Bases

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