An Alexander-type duality for valuations

TL;DR

This paper establishes a new Alexander-type duality for valuations on specific subcomplexes of polyhedral boundaries, extending classical results and providing a generalized theorem with topological insights.

Contribution

It introduces a novel duality for valuations on boundary subcomplexes, generalizing Brion's theorem and simplifying prior results by Stanley and Miller-Reiner.

Findings

Proved an Alexander-type duality for valuations on boundary subcomplexes.

Generalized Brion's theorem to a relative setting.

Analyzed the topology of subcomplexes satisfying the duality.

Abstract

We prove an Alexander-type duality for valuations for certain subcomplexes in the boundary of polyhedra. These strengthen and simplify results of Stanley (1974) and Miller-Reiner (2005). We give a generalization of Brion's theorem for this relative situation and we discuss the topology of the possible subcomplexes for which the duality relation holds.