Coxeter elements and Coxeter plane: A programming guide

Abstract In a finite Coxeter group, the product of its generating elements in any given sequence is called a Coxeter element. The Coxeter element acts on a two-dimensional plane P as a rotation, and the rotation’s order is precisely the order of the Coxeter element itself. When projecting the root system onto P, the result is a beautifully symmetric pattern that exhibits rotational symmetry. This paper takes the exceptional Lie algebra E 8 as an example to demonstrate how to compute such a projection using the Python programming language. We also discuss some properties of Coxeter elements, such as their conjugacy, orders, characteristic polynomials, and eigenvalues. This contribution opens up new avenues for a deeper understanding of Lie algebras and Coxeter groups, enriching the reader’s comprehension of these beautiful structures.