Adapted sequences and polyhedral realizations of crystal bases for highest weight modules
Adapted sequences and polyhedral realizations of crystal bases for highest weight modules
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针对最高重量模块的晶体基座的调整序列和多面体实现
DOI：
10.1016/j.jalgebra.2021.01.016
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发表时间：
2021
影响因子：
0.9
通讯作者：
Toshiki Nakashima with Yuki Kanakubo
中科院分区：
文献类型：
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作者：
Ryoichi Kuwano;Makoto Hino;Norihito Nagata;Kazuya Nagata;Tsuyoshi Tokunaga;Toshihiko Koga;Nathan Hagen;Yukitoshi Otani;Toshiki Nakashima with Yuki Kanakubo
Nakashima (1999) [13] 引入了 U q (g) 可积最高权模块的晶体基的多面体实现，它将晶体基描述为由线性不等式组给出的无限 Z 晶格 Z∞ 中的晶格点集，其中 g 是可对称的 Kac-Moody 李代数。为了构造多面体实现，我们需要根据简单根的索引确定无限序列 ι。如果 (ι, λ)（λ：主导积分权重）对满足“充足”条件，则有一些过程可以计算线性不等式组。在本文中，我们证明，如果 ι 是一个适应序列（在我们的论文 [5] 中定义），那么在 g 是经典李代数的情况下，对 (ι, λ) 满足任何主导积分权重 λ 的充分条件。此外，我们还揭示了与任意适应序列 ι 相关的晶体基 B (λ) 的多面体实现的显式形式。作为一个应用，我们将给出晶体基 B (∞) 上函数 ε i⁎ 的组合描述。
The polyhedral realizations for crystal bases of the integrable highest weight modules of U q (g) have been introduced in Nakashima (1999)[13], which describe the crystal bases as sets of lattice points in the infinite Z-lattice Z∞ given by some system of linear inequalities, where g is a symmetrizable Kac-Moody Lie algebra. To construct the polyhedral realization, we need to fix an infinite sequence ι from the indices of the simple roots. If the pair (ι, λ)(λ: a dominant integral weight) satisfies the ‘ample’condition then there are some procedure to calculate the sets of linear inequalities. In this article, we show that if ι is an adapted sequence (defined in our paper [5]) then the pair (ι, λ) satisfies the ample condition for any dominant integral weight λ in the case g is a classical Lie algebra. Furthermore, we reveal the explicit forms of the polyhedral realizations of the crystal bases B (λ) associated with arbitrary adapted sequences ι in terms of column tableaux. As an application, we will give a combinatorial description of the function ε i⁎ on the crystal base B (∞).