giove91 / affine-coxeter

Hyperbolic elements in affine Coxeter groups

This program checks Lemma 3.21 of [PS19] for the exceptional affine Coxeter groups F4, E6, E7, E8 (the case G2 can be easily checked by hand). The lemma is checked for all hyperbolic elements u in [1,w], without the irreducibility hypothesis.

Requirements: SageMath 8.8 and Python 2.7.

Usage

sage check_hyperbolic.sage F4
sage check_hyperbolic.sage E6
sage check_hyperbolic.sage E7
sage check_hyperbolic.sage E8

The optional argument -v can be added to print more information.

The case C is also implemented. For example:

sage check_hyperbolic.sage C2

Bibliography

[Arm09] D. Armstrong, Generalized noncrossing partitions and combinatorics of Coxeter groups, Memoirs of the American Mathematical Society (2009).

[Hum92] J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press (1992).

[MS17] J. McCammond and R. Sulway, Artin groups of Euclidean type, Inventiones Mathematicae 210(1), 231-282 (2017).

[PS19] G. Paolini and M. Salvetti, Proof of the K(π,1) conjecture for affine Artin groups, arXiv preprint 1907.11795 (2019).