嵌套多面体与非晶体型Coxeter群的轨道指数
Nested polyhedra and indices of orbits of Coxeter groups of non-crystallographic type
摘要：
在非晶体学有限反射群$H_2$、$H_3$和$H_4$的背景下，考虑了单李代数有限维表示的不变量，如偶度指数和反常数.使用一个代表轨道替换，指数的定义和性质制定为个别轨道的检查组。确定了$k$轨道张量积的二阶和四阶指数。使用非晶体学Coxeter群的分支规则，嵌入指数的定义类似于表示的Dynkin指数。此外，由于指数的定义可以应用于任何非晶体学类型的轨道，算法允许搜索包含在任何考虑的一个较小半径的轨道的Coxeter群$H_2$和$H_3$。嵌套多面体的几何结构的例子。
英文摘要：
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.
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