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

TL;DR

The paper develops explicit, efficient formulas for structure constants in multiply-laced root systems $B_n$, $C_n$, and $F_4$ by leveraging Carter’s special and extraspecial root pairs and the quartet framework. It classifies quartets into mono and simple types, showing that $B_n$ is of Vavilov type (all quartets mono and simple) and that $C_n$ quartets are always simple with a length-ratio parameter $\varphi$, while $F_4$ contains both simple and non-simple quartets. The results yield reduced computations of squared-length terms and provide concrete final formulas for $N(r,s)$, along with tables and a Python implementation to facilitate practical calculations. This advancement streamlines the calculation of structure constants in these non-simply-laced cases and supports broader applications in representation theory and Lie algebra computations. The work integrates Carter’s theorems, Chevalley basis theory, and inductive algorithms to deliver both theoretical insights and actionable computational tools.

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 $\{r_1, s_1\}$.

Theorems & Definitions (25)

• Proposition 1.1
• Corollary 1.2
• Corollary 1.3
• Theorem 2.1: Chevalley, 1955
• Theorem 2.2: Carter, 1972
• Lemma 2.3
• Lemma 2.4
• Remark 2.5
• Lemma 3.1: H72, Ch.10.2
• Corollary 3.2