论文标题
Ariki- Koike代数和Rogers- Ramanujan类型分区
The Ariki--Koike algebras and Rogers--Ramanujan type partitions
论文作者
论文摘要
In 2000, Ariki and Mathas showed that the simple modules of the Ariki--Koike algebras $\mathcal{H}_{\mathbb{C},q;Q_1,\ldots, Q_m}\big(G(m, 1, n)\big)$ (when the parameters are roots of unity and $q\neq 1$) are labeled by the所谓的Kleshchev多阶。与Ariki的分类定理一起，通过利用Weyl-kac角色公式，使Ariki和Mathas能够获得Kleshchev多阶数的生成函数。在本文中，我们为$ q = -1 $ case重新访问此生成功能。此$ q = -1 $ case特别有趣，因为相应的kleshchev多阶层与广义的罗杰斯（Ramanujan）类型分区非常紧密，当$ q_1 = \ cdots = q_a = q_a = -1 $和$ q_ {a+1} = \ cdots = \ cdots = q_m = q_m = 1 $。基于此连接，我们为$ q = q_1 = \ cdots q_a = -1 $和$ q_ {a+1} = \ cdots = q_m = 1 $提供了Ariki和Mathas结果的分析证明。我们的第二个目标是研究Ariki的简单模块 - 固定块中的koike代数。众所周知，固定块中的这些简单模块由具有固定分区残基统计量的Kleshchev多解析标记。该分区统计数据还在Berkovich，Garvan和UNCU的作品中进行了研究。使用他们的结果，当$ M = 2 $时，我们提供两个双变量生成功能身份。
In 2000, Ariki and Mathas showed that the simple modules of the Ariki--Koike algebras $\mathcal{H}_{\mathbb{C},q;Q_1,\ldots, Q_m}\big(G(m, 1, n)\big)$ (when the parameters are roots of unity and $q\neq 1$) are labeled by the so-called Kleshchev multipartitions. This together with Ariki's categorification theorem enabled Ariki and Mathas to obtain the generating function for the number of Kleshchev multipartitions by making use of the Weyl--Kac character formula. In this paper, we revisit this generating function for the $q=-1$ case. This $q=-1$ case is particularly interesting, for the corresponding Kleshchev multipartitions have a very close connection to generalized Rogers--Ramanujan type partitions when $Q_1=\cdots=Q_a=-1$ and $Q_{a+1}=\cdots =Q_m =1$. Based on this connection, we provide an analytic proof of the result of Ariki and Mathas for $q=Q_1=\cdots Q_a=-1$ and $Q_{a+1}=\cdots =Q_m =1$. Our second objective is to investigate simple modules of the Ariki--Koike algebra in a fixed block. It is known that these simple modules in a fixed block are labeled by the Kleshchev multiparitions with a fixed partition residue statistic. This partition statistic is also studied in the works of Berkovich, Garvan, and Uncu. Employing their results, we provide two bivariate generating function identities when $m=2$.
