发表机构
Tutte Institute of Mathematics and Computing（图特数学与计算研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明秩2下仿射MV多面体与约化双Bruhat胞腔的非负热带点一一对应，推广了有限型结果，并因反射对称性同样适用于上仿射MV多面体。
AI 中文摘要
当$G$是复约化代数群时，MV多面体与$G$的幂单群的非负热带点一一对应。本文对秩2仿射MV多面体的某些子类证明了类似定理。对于Kac-Moody群$\widehat{SL_2}$，一个仿射MV多面体分裂为三个子多面体：下多面体、中多面体和上多面体。下多面体是有限型多面体的自然推广，其最高顶点由任意Weyl元素标记。我们将有限型MV多面体这一子类的已知结果推广到秩2的下仿射MV多面体情形。我们证明，对于仿射Weyl群中的元素$w$，以$w$为最高顶点的下仿射MV多面体类与标记为$w^{-1}$的约化双Bruhat胞腔的非负热带点一一对应。为此，我们描述了秩2仿射MV多面体的BZ数据，并证明了某些广义子式函数满足下多面体的条件。由于任何上多面体都是某个下多面体的反射，类似结果也将对秩2的上仿射MV多面体类成立。
英文摘要
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 $\widehat{SL_2}$, 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.
Comments23 pages, 5 figures
