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

TL;DR

This paper investigates the structure of root systems in Lie algebras, introducing the concept of quartets involving special and extraspecial root pairs to simplify the calculation of structure constants, especially in types Bn and Cn.

Contribution

It provides new methods to compute structure constants in multiply-laced root systems by utilizing quartets, reducing computational complexity for types Bn and Cn.

Findings

Avoids calculating 6 squares of lengths in Bn case

Formula for Bn matches simply-laced case

Reduces to calculating 4 squares in Cn case

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 $\{r_1, s_1\}$.