论文标题
某些有限组的麦凯箭和卢斯蒂格代数
McKay quivers and Lusztig algebras of some finite groups
论文作者
论文摘要
我们对McKay Quiver $γ（g）$和偏斜式环$ a*g $感兴趣，其中$ g $是$ \ mathrm {gl}（v）$的有限子组，其中$ v $是field $ k $的有限尺寸矢量空间，而$ a $ a $ a $ a $ kg $ kg $ -Algeberbra。这些偏斜的组戒指出现在Auslander版的McKay通信中。 在本文的第一部分中，我们考虑了复杂的反射组$ g \ subseteq \ mathrm {gl}（v）$，并找到一种组合方法，利用年轻图来构建$ g（r，p，n）$组的McKay Quivers。我们首先查看$ g（1,1，n）$，这是对称组$ s_n $的同构，其次是$ g（r，1，n）$，$ r> 1 $。然后，使用Clifford理论，我们可以确定任何$ G（R，P，N）$的McKay颤抖，因此对于所有有限的不可删除的复杂反射组，以达到有限的许多例外。 在本文的第二部分中，我们考虑了一种对任意有限群体的McKay Quivers的更概念性方法：我们定义了有限组的$ g \ subseteq \ subseteq \ subseteq \ mathrm {gl}（gl}（v）$的lusztig代数$ \ widetilde a（g）$，这是Morita equivalent the Morita equivalents y Morita quivalent the Morita equivalent the ske group $ a*g $ a*g $ a*g $。此描述使我们将基本代数莫里塔（Morita）的嵌入等同于$ a*g $，超过$ a $的矩阵代数。
We are interested in the McKay quiver $Γ(G)$ and skew group rings $A*G$, where $G$ is a finite subgroup of $\mathrm{GL}(V)$, where $V$ is a finite dimensional vector space over a field $K$, and $A$ is a $K-G$-algebra. These skew group rings appear in Auslander's version of the McKay correspondence. In the first part of this paper we consider complex reflection groups $G \subseteq \mathrm{GL}(V)$ and find a combinatorial method, making use of Young diagrams, to construct the McKay quivers for the groups $G(r,p,n)$. We first look at the case $G(1,1,n)$, which is isomorphic to the symmetric group $S_n$, followed by $G(r,1,n)$ for $r >1$. Then, using Clifford theory, we can determine the McKay quiver for any $G(r,p,n)$ and thus for all finite irreducible complex reflection groups up to finitely many exceptions. In the second part of the paper we consider a more conceptual approach to McKay quivers of arbitrary finite groups: we define the Lusztig algebra $\widetilde A(G)$ of a finite group $G \subseteq \mathrm{GL}(V)$, which is Morita equivalent to the skew group ring $A*G$. This description gives us an embedding of the basic algebra Morita equivalent to $A*G$ into a matrix algebra over $A$.
