6 research outputs found
Zonotopal algebra
A wealth of geometric and combinatorial properties of a given linear
endomorphism of is captured in the study of its associated zonotope
, and, by duality, its associated hyperplane arrangement .
This well-known line of study is particularly interesting in case n\eqbd\rank
X \ll N. We enhance this study to an algebraic level, and associate with
three algebraic structures, referred herein as {\it external, central, and
internal.} Each algebraic structure is given in terms of a pair of homogeneous
polynomial ideals in variables that are dual to each other: one encodes
properties of the arrangement , while the other encodes by duality
properties of the zonotope . The algebraic structures are defined purely
in terms of the combinatorial structure of , but are subsequently proved to
be equally obtainable by applying suitable algebro-analytic operations to
either of or . The theory is universal in the sense that it
requires no assumptions on the map (the only exception being that the
algebro-analytic operations on yield sought-for results only in case
is unimodular), and provides new tools that can be used in enumerative
combinatorics, graph theory, representation theory, polytope geometry, and
approximation theory.Comment: 44 pages; updated to reflect referees' remarks and the developments
in the area since the article first appeared on the arXi