论文标题
嵌套的多面体和非晶体学类型的Coxeter组的轨道指标
Nested polyhedra and indices of orbits of Coxeter groups of non-crystallographic type
论文作者
论文摘要
在非结晶有限反射组$ h_2 $，$ h_3 $和$ h_4 $的背景下，考虑了简单谎言代数的有限维表示的不变式（例如级别索引和异常数字）。使用表示轨道替换，该索引的定义和属性是针对所检查组的各个轨道制定的。确定了$ k $轨道的张量产品的两个和四个订单指数。使用非晶体构型Coxeter组的分支规则，嵌入索引的定义与表示形式的Dynkin索引相似。此外，由于该索引的定义可以应用于非结晶类型的任何轨道，因此该算法允许搜索为Coxeter组$ H_2 $和$ H_3 $提供任何所考虑的一个较小半径的轨道。嵌套多面体的几何结构被例证。
The invariants of finite-dimensional representations of simple Lie algebras, such as even-degree indices and anomaly numbers, are considered in the context of the non-crystallographic finite reflection groups $H_2$, $H_3$ and $H_4$. Using a representation-orbit replacement, the definitions and properties of the indices are formulated for individual orbits of the examined groups. The indices of orders two and four of the tensor product of $k$ orbits are determined. Using the branching rules for the non-crystallographic Coxeter groups, the embedding index is defined similarly to the Dynkin index of a representation. Moreover, since the definition of the indices can be applied to any orbit of non-crystallographic type, the algorithm allowing to search for the orbits of smaller radii contained within any considered one is presented for the Coxeter groups $H_2$ and $H_3$. The geometrical structures of nested polytopes are exemplified.
