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

2024/09/20 by Rafael Stekolshchik, Stekolshchik, Rafael
Chemistry · Materials Science · #17B05 #17B22 #FOS: Mathematics #Liquid Crystal Research Advancements #Molecular spectroscopy and chirality #Representation Theory (math.RT) #Spectroscopy and Chemometric Analyses

paper · pdf · doi:10.48550/arxiv.2409.13552

openalex publication_date 2024/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

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 \r₁, s₁\ be the extraspecial pair of roots corresponding to \r, s\. Consider the ordered set \r₁, r, s, s₁\, 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 \r₁, s₁\.

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