An Intersection Product for the Polytope Algebra
Combinatorics
2025-05-12 v2 Metric Geometry
Abstract
We introduce a new multiplication for the polytope algebra, defined via the intersection of polytopes. After establishing the foundational properties of this intersection product, we investigate finite-dimensional subalgebras that arise naturally from this construction. These subalgebras can be regarded as volumetric analogues of the graded M\"obius algebra, which appears in the context of the Dowling-Wilson conjecture. We conjecture that they also satisfy the injective hard Lefschetz property and the Hodge-Riemann relations, and we prove these in degree one.
Cite
@article{arxiv.2504.16678,
title = {An Intersection Product for the Polytope Algebra},
author = {Thomas Wannerer},
journal= {arXiv preprint arXiv:2504.16678},
year = {2025}
}