Regular dyadic triangulations of delta-matroid polytopes

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Abstract

Backman and Liu proved that every integral generalized permutohedron of type $A$, and in particular every matroid base polytope, admits a regular unimodular triangulation. The analogous statement fails in type $B$: the delta-matroid simplex $conv\0, e_1+e_2, e_1+e_3, e_2+e_3$ has normalized volume $2$ and no lattice points other than its vertices, so it has no unimodular triangulation. We show moreover that, up to the natural symmetries of the $0/1$ cube and deletion of constant coordinates, it is the unique non-unimodular delta-matroid polytope that is a simplex. We prove instead that every delta-matroid polytope admits a regular dyadic triangulation, meaning a lattice triangulation whose maximal simplices have normalized volumes that are powers of two. More generally, every integral type $B$ generalized permutohedron admits such a triangulation. The main lattice-theoretic ingredient is that the type $B$ root configuration forms a totally dyadic system, a $2$-local analogue of total unimodularity. As a consequence, these polytopes satisfy a dyadic version of the integer decomposition property. In each dimension the corresponding exponent can be chosen uniformly, even though ordinary integer decomposition can fail for delta-matroid polytopes.

Citation

Vallée, Mathieu. 2026. “Regular dyadic triangulations of delta-matroid polytopes.” https://arxiv.org/abs/2609.18331

@misc{vallee_regular_2026,
  author = {Vallée, Mathieu},
  year = {2026},
  title = {Regular dyadic triangulations of delta-matroid polytopes},
  doi = {10.48550/arXiv.2609.18331},
  url = {http://arxiv.org/abs/2609.18331},
  note = {arXiv:2609.18331 [math.CO]},
  publisher = {arXiv},
}