Extraspecial pairs in the multiply-laced root systems and calculating structure constants

Abstract

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 $\{r_1, s_1\}$.