Adapted sequence for polyhedral realization of crystal bases
Adapted sequence for polyhedral realization of crystal bases
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DOI：
10.1080/00927872.2020.1770274
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发表时间：
2019-04
影响因子：
0.7
通讯作者：
Yuki Kanakubo;Toshiki Nakashima
中科院分区：
文献类型：
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作者：
Yuki Kanakubo;Toshiki Nakashima
Zelevinsky和Nakashima提出了晶体基的多面体实现，将晶体基描述为无限晶格中的多面体凸锥。根据这一点，有一定的一组线性函数定义了一个多面体凸锥，并在“正性条件”下对ι，它已被证明，多面体凸锥同构的晶体基地，以确定为一个给定的正性条件，我们需要获得的线性函数集的整体特征，这需要，在一般情况下，一堆明确的计算。在这篇文章中，我们引入了适应序列的概念，并证明了如果ι是一个适应序列，那么经典李代数的正性条件成立。此外，我们揭示了显式形式的多面体实现与任意适应序列i列tableaux。
Abstract The polyhedral realization of crystal base has been introduced by Zelevinsky and Nakashima, which describe the crystal base as a polyhedral convex cone in the infinite -lattice To construct the polyhedral realization, we need to fix an infinite sequence ι from the indices of the simple roots. According to this one has certain set of linear functions defining a polyhedral convex cone and under the “positivity condition” on ι, and it has been shown that the polyhedral convex cone is isomorphic to the crystal base To confirm the positivity condition for a given ι, we need to obtain the whole feature of the set of linear functions, which requires, in general, a bunch of explicit calculations. In this article, we introduce the notion of the adapted sequence and show that if ι is an adapted sequence, then the positivity condition holds for classical Lie algebras. Furthermore, we reveal the explicit forms of the polyhedral realizations associated with arbitrary adapted sequences ι in terms of column tableaux.