任意根系的拟晶体和次胚幺半群的相关推广
Quasi-crystals for arbitrary root systems and associated generalizations of the hypoplactic monoid
摘要：
下胚么半群是由Krob和Thibon通过一个表示、拟带状图和插入算法引入的。正如Kashiwara晶体丰富了plastic幺半群的结构，并允许其推广，本文的第一和第三作者介绍了一种构造hypoplastic幺半群的方法，通过确定从与一般线性李代数相关联的晶体图导出的准晶体图中的顶点。虽然这种构造是基于Kashiwara的工作，但它不能扩展到其他晶体图，因为类似的词上的准Kashiwara算子不允许递归定义。本文件讨论这些问题。介绍了准晶的一般概念，研究了准晶的性质及其与晶体的关系。准晶体的组合研究，然后通过将准晶体图关联到每个准晶体，这对于类非正规准晶体的结果是一一对应的。为了用准晶体来模拟亚塑性幺半群的二元运算，引入了准晶体的准张量积的概念，并给出了一种类似于晶体的张量积的符号规则的计算方法.这个框架允许推广的经典hypoplastic幺半群的hypoplastic幺半群的家庭与各种简单的李代数。准晶体结构，然后用来建立与辛李代数的hypoplastic幺半群的代数性质。
英文摘要：
The hypoplactic monoid was introduced by Krob and Thibon through a presentation and through quasi-ribbon tableaux and an insertion algorithm. Just as Kashiwara crystals enriched the structure of the plactic monoid and allowed its generalization, the first and third authors of this paper introduced a construction of the hypoplactic monoid by identifying vertices in a quasi-crystal graph derived from the crystal graph associated to the general linear Lie algebra. Although this construction is based on Kashiwara's work, it cannot be extended to other crystal graphs, since the analogous quasi-Kashiwara operators on words do not admit a recursive definition. This paper addresses these issues. A general notion of quasi-crystal is introduced, followed by a study of its properties and relation with crystals. A combinatorial study of quasi-crystals is then made by associating a quasi-crystal graph to each quasi-crystal, which for the class of seminormal quasi-crystals results in a one-to-one correspondence. To model the binary operation of the hypoplactic monoid by quasi-crystals, a notion of quasi-tensor product of quasi-crystals is introduced, along with a combinatorial way of computing it similar to the signature rule for the tensor product of crystals. This framework allows the generalization of the classical hypoplactic monoid to a family of hypoplactic monoids associated to the various simple Lie algebras. The quasi-crystal structure is then used to establish algebraic properties of the hypoplactic monoid associated to the symplectic Lie algebra.
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