论文标题
最高重量模块的晶体底座的适应序列和多面体实现
Adapted Sequences and Polyhedral Realizations of Crystal Bases for highest weight modules
论文作者
论文摘要
在（[T.Nakashima，J。Algebra，vol.219，o.2，（1999））中引入了$ u_q（\ mathfrak {g}）$的晶体基础的多面化实现，（\ nakashima，vol.219） $ \ mathbb {z}^{\ infty} $由某些线性不平等系统给出，其中$ \ mathfrak {g} $是一种同样的可kac-moody lie代数。为了构建多面体实现，我们需要从简单根部的索引中修复无限序列$ b。如果这对（$ i $，$λ$）（$λ$：主体积分重量）满足“充足”条件，那么有一些程序可以计算线性不平等的集合。 在本文中，我们表明，如果$ b $是一个改编的序列（在我们的论文中定义[Y.Kanakubo，T.Nakashima，T.Nakashima，arxiv：1904.10919]），那么这对（$ $ $，$ c，$λ$）可以满足任何主要的整体体重$λ$ case $λ$的lie lie utive $λ$ \ nie classe al classe al a al a al a al a al a a al a a a g} $ a a a g} $ a a a g} $ a a a g} $ a a a g} $ a。此外，我们揭示了与任意改编的序列相关的水晶基础$ b（λ）$的明确形式。作为一个应用程序，我们将在Crystal Base $ b（\ infty）$上提供功能$ \ varepsilon_i^*$的组合描述。
The polyhedral realizations for crystal bases of the integrable highest weight modules of $U_q(\mathfrak{g})$ have been introduced in ([T.Nakashima, J. Algebra, vol.219, no. 2, (1999)]), which describe the crystal bases as sets of lattice points in the infinite $\mathbb{Z}$-lattice $\mathbb{Z}^{\infty}$ given by some system of linear inequalities, where $\mathfrak{g}$ 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 [Y.Kanakubo, T.Nakashima, arXiv:1904.10919]) then the pair ($ι$, $λ$) satisfies the ample condition for any dominant integral weight $λ$ in the case $\mathfrak{g}$ 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 $\varepsilon_i^*$ on the crystal base $B(\infty)$.
