多重边根系中的超特殊对与结构常数计算
Extraspecial pairs in the multiply-laced root systems and calculating structure constants
AI总结：
本文研究多重边根系中用于计算结构常数的特殊对与超特殊对，通过四元组分析，发现 B_n 和 C_n 情形可简化计算，并给出与单边情形的关系。
摘要：
Carter 引入了根的特殊对和超特殊对的概念，用于计算结构常数，[Ca72]。设 {r, s} 为待求结构常数 N(r,s) 的特殊根对，并设 {r_1, s_1} 为对应于 {r, s} 的超特殊根对。考虑有序集合 {r_1, r, s, s_1}，我们称这样的集合为四元组。通过研究不同的四元组，我们得以更深入地了解根系的内部结构。结果表明，对于 B_n 情形，我们可以避免在计算结构常数的公式中计算 6 个长度平方。B_n 的计算公式与单边情形一致。对于 C_n 情形，可以避免计算 4 个长度平方。C_n 的计算公式与单边情形不同，相差某个参数，该参数对于所有具有给定超特殊对 {r_1, s_1} 的根对 {r, s} 是固定的。
英文摘要：
The notions of special and extraspecial pairs of roots were introduced by Carter for calculating structure constants, [Ca72]. Let $\{r, s\}$ be a special pair of roots for which the structure constant $N(r,s)$ is sought, and let $\{r_1, s_1\}$ be the extraspecial pair of roots corresponding to $\{r, s\}$. Consider the ordered set $\{r_1, r, s, s_1\}$, we will call such a set a quartet. By studying the different quartets, we gain additional insight into the internal structure of the root system. It is shown that for the case $B_n$ we can avoid finding $6$ squares of lengths in the formula for calculating the structure constants. The calculation formula for $B_n$ coincides with the formula for the simply-laced case. For the case $C_n$, it is possible to avoid the calculation of $4$ squares of lengths. The calculation formula for $C_n$ differs from simply-laced case by some parameter, which is fixed for all pairs $\{r, s\}$ with given extraspecial pair $\{r_1, s_1\}$.
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