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PreprintCC BY-NC-ND 4.0arXiv

Lower affine MV polytopes of rank 2

  • Kathlyn Dykes
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Abstract

When G is a complex reductive algebraic group, MV polytopes are in bijection with the non-negative tropical points of the unipotent group of G. In this paper, we prove a similar theorem for certain subclasses of rank 2 affine MV polytopes. For the Kac-Moody group SL2̂, an affine MV polytopes splits into three subpolytopes: a lower, a middle and an upper polytope. The lower polytopes are natural generalizations of finite-type polytopes with highest vertex labelled by an arbitrary Weyl element. We extend the known results from this subclass of finite-type MV polytopes to the case of lower affine MV polytopes of rank 2. We prove that for an element w of the affine Weyl group, the class of lower affine MV polytopes with highest vertex w are in bijection with the non-negative tropical points of the reduced double Bruhat cell labelled by w−1. To do this, we describe the BZ data of a rank 2 affine MV polytope and show that certain generalized minor functions satisfy the conditions of a lower polytope. As any upper polytope is a reflection of some lower polytope, a analogous result will hold for the class of upper affine MV polytopes of rank 2.

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Cite

BibTeXRIS
BibTeX
@misc{dykes2026lower,
  title = {{Lower affine MV polytopes of rank 2}},
  author = {Kathlyn Dykes},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39771},
  eprint = {2609.39771},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39771},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Lower affine MV polytopes of rank 2
AU  - Kathlyn Dykes
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39771
UR  - https://arxiv.org/abs/2609.39771
AB  - When G is a complex reductive algebraic group, MV polytopes are in bijection with the non-negative tropical points of the unipotent group of G. In this paper, we prove a similar theorem for certain subclasses of rank 2 affine MV polytopes. For the Kac-Moody group SL_2̂, an affine MV polytopes splits into three subpolytopes: a lower, a middle and an upper polytope. The lower polytopes are natural generalizations of finite-type polytopes with highest vertex labelled by an arbitrary Weyl element. We extend the known results from this subclass of finite-type MV polytopes to the case of lower affine MV polytopes of rank 2. We prove that for an element w of the affine Weyl group, the class of lower affine MV polytopes with highest vertex w are in bijection with the non-negative tropical points of the reduced double Bruhat cell labelled by w^−1. To do this, we describe the BZ data of a rank 2 affine MV polytope and show that certain generalized minor functions satisfy the conditions of a lower polytope. As any upper polytope is a reflection of some lower polytope, a analogous result will hold for the class of upper affine MV polytopes of rank 2.
ER  -