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A cubulation of the associahedron using a torus-equivariant degeneration of the brick variety

  • Raj Gandhi
  • Gabe Udell
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Abstract

We describe a mixed subdivision of the (n−1)-dimensional Loday associahedron into (n−1)! maximal cells which are combinatorial cubes, and we show that the face poset of this subdivision is isomorphic to the face poset of the subdivision of the 1× 2× dotsb × (n−1) box into unit cubes. More generally, we describe mixed subdivisions of generalized associahedra into pieces which are Minkowski sums of Bruhat interval polytopes. Our subdivisions arise from a torus-equivariant embedded (in a product of flag varieties) degeneration of brick varieties into reduced unions of products of Richardson varieties. The subdivision of Loday's associahedron allows us to `embed' the 1-skeleton of the permutahedron in the boundary of the associahedron.

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Cite

BibTeXRIS
BibTeX
@misc{gandhi2026cubulation,
  title = {{A cubulation of the associahedron using a torus-equivariant degeneration of the brick variety}},
  author = {Raj Gandhi and Gabe Udell},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23004},
  eprint = {2610.23004},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23004},
}
RIS
TY  - UNPB
TI  - A cubulation of the associahedron using a torus-equivariant degeneration of the brick variety
AU  - Raj Gandhi
AU  - Gabe Udell
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23004
UR  - https://arxiv.org/abs/2610.23004
AB  - We describe a mixed subdivision of the (n−1)-dimensional Loday associahedron into (n−1)! maximal cells which are combinatorial cubes, and we show that the face poset of this subdivision is isomorphic to the face poset of the subdivision of the 1× 2× dotsb × (n−1) box into unit cubes. More generally, we describe mixed subdivisions of generalized associahedra into pieces which are Minkowski sums of Bruhat interval polytopes. Our subdivisions arise from a torus-equivariant embedded (in a product of flag varieties) degeneration of brick varieties into reduced unions of products of Richardson varieties. The subdivision of Loday's associahedron allows us to `embed' the 1-skeleton of the permutahedron in the boundary of the associahedron.
ER  -