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July 10, 20260 citationsOpen Access

Atlas of Weyl Chambers of E₈: First Algorithmic Classification of Fundamental Weights

Key Points

To create an algorithmic atlas of Weyl chambers for the exceptional Lie algebra E₈ and classify fundamental weights.
Complete scan of all 120 positive roots to identify unique Weyl chambers.
Calculation of fundamental weights and dimensions of fundamental representations for each chamber.
Inclusion of all sources and datasets in LaTeX, Python, and JSON formats.
Identified exactly 8 unique Weyl chambers, all sharing the same fundamental dimensions: {248, 3875, 30380, 147250, 2450240, 6696000, 146325270, 6899079264}.
Confirmed identical Cartan matrix invariants (trace=16, determinant=1) for all chambers.
No prime fundamental dimensions were found across the chambers.

Abstract

First algorithmic atlas of Weyl chambers for the exceptional Lie algebra E₈. Complete scan of all 120 positive roots as seeds yields exactly 8 unique Weyl chambers. For each chamber, fundamental weights and dimensions of fundamental representations are computed. All 8 chambers have the identical set of 8 fundamental dimensions: 248, 3875, 30380, 147250, 2450240, 6696000, 146325270, 6899079264. Cartan matrix invariants (trace=16, determinant=1) are identical for all chambers, confirming they belong to the same E₈ algebra. No prime fundamental dimensions are found. Includes: LaTeX sources (RU + EN), Python source code (RU + EN), JSON dataset with full atlas data. --------- Первый алгоритмический атлас камер Вейля исключительной алгебры Ли E₈. Полный перебор 120 положительных корней дал ровно 8 уникальных камер. Для каждой вычислены фундаментальные веса и размерности. Все 8 камер имеют одинаковый набор размерностей. Инварианты Картана (след=16, определитель=1) одинаковы. Включает: LaTeX (RU+EN), Python (RU+EN), JSON с полными данными.

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