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Berenstein-Zelevinsky polytopes for type G2

  • Thomas Magnuson
  • Tyrrell McAllister
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Abstract

In 2001, Berenstein and Zelevinsky presented a polyhedral model for the problem of computing tensor-product multiplicities for semi\-simple Lie algebras. We provide explicit linear equations and inequalities for Berenstein--Zelevinsky (BZ) polytopes of type-G2 Lie algebras. We prove that the denominator of a type-G2 BZ~polytope divides $180$, and that this bound is sharp, while the quasi-period of the Ehrhart quasipolynomial of such a polytope is always a divisor of $6$. We also show that BZ~polytopes of type-G2 Lie algebras can fail to satisfy Ehrhart positivity, in contrast to a conjectured property of BZ~polytopes for all classical types.

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Cite

BibTeXRIS
BibTeX
@misc{magnuson2026berensteinzelevinsky,
  title = {{Berenstein-Zelevinsky polytopes for type $\mathrmG_2$}},
  author = {Thomas Magnuson and Tyrrell McAllister},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.42038},
  eprint = {2610.42038},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.42038},
}
RIS
TY  - UNPB
TI  - Berenstein-Zelevinsky polytopes for type G_2
AU  - Thomas Magnuson
AU  - Tyrrell McAllister
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.42038
UR  - https://arxiv.org/abs/2610.42038
AB  - In 2001, Berenstein and Zelevinsky presented a polyhedral model for the problem of computing tensor-product multiplicities for semi\-simple Lie algebras. We provide explicit linear equations and inequalities for Berenstein--Zelevinsky (BZ) polytopes of type-G_2 Lie algebras. We prove that the denominator of a type-G_2 BZ~polytope divides $180$, and that this bound is sharp, while the quasi-period of the Ehrhart quasipolynomial of such a polytope is always a divisor of $6$. We also show that BZ~polytopes of type-G_2 Lie algebras can fail to satisfy Ehrhart positivity, in contrast to a conjectured property of BZ~polytopes for all classical types.
ER  -