Combinatorial realizations of crystals via torus actions on quiver varieties
Combinatorial realizations of crystals via torus actions on quiver varieties
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通过对箭袋品种的环面作用实现晶体的组合
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发表时间：
2012
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通讯作者：
P. Tingley
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作者：
Steven V. Sam;P. Tingley
设V（λ）是对称Kac-Moody代数的最高权表示，设B（λ）是它的晶体。B（λ）的几何实现是利用Nakajima的颤抖变体。在许多特殊情况下，我们也可以用初等组合方法来实现B（λ）。本文研究了一种从几何图中提取组合实现的一般方法：利用莫尔斯理论，利用环面作用下不动点子簇的连通分量来索引不可约分量。然后我们讨论$widehat{mathfrak{sl}}_{n}$的情况，其中不动点组件只是点，并且自然地被多分区索引。在我们的结构中有一些选择，导致每个最高重量晶体的组合实现家族。以B（Λ0）为例，我们恢复了法耶斯最近构建的一系列实现。这为Fayers的结果提供了更概念性的证明，并将其推广到更高层次的晶体。我们还讨论了与中岛单项式晶体的关系。
Let V(λ) be a highest-weight representation of a symmetric Kac–Moody algebra, and let B(λ) be its crystal. There is a geometric realization of B(λ) using Nakajima’s quiver varieties. In many particular cases one can also realize B(λ) by elementary combinatorial methods. Here we study a general method of extracting combinatorial realizations from the geometric picture: we use Morse theory to index the irreducible components by connected components of the subvariety of fixed points for a certain torus action. We then discuss the case of $widehat{mathfrak{sl}}_{n}$, where the fixed point components are just points, and are naturally indexed by multi-partitions. There is some choice in our construction, leading to a family of combinatorial realizations for each highest-weight crystal. In the case of B(Λ0) we recover a family of realizations which was recently constructed by Fayers. This gives a more conceptual proof of Fayers’ result as well as a generalization to higher level crystals. We also discuss a relationship with Nakajima’s monomial crystal.