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October 5, 2025Open Access

Canonical Reduction Systems in Artin-Tits groups of spherical type

Puntos clave

The work establishes the canonical reduction system of elements in Artin-Tits groups, extending similar concepts in braid groups.
A comprehensive algorithm is introduced alongside improvements for existing braid group algorithms, optimizing computation efficiency.
The research demonstrates that centralizers of positive powers create a periodic sequence, adding depth to the understanding of Artin-Tits groups.
Findings highlight important properties shared between Artin-Tits groups and braid groups, expanding their mathematical relationship.

Resumen

We introduce the canonical reduction system of an element in an Artin-Tits group of spherical type, which generalizes the similar notion for braids (and mapping classes) introduced by Birman, Lubotzky and McCarthy. We show its basic properties, which coincide with those satisfied in braid groups, and we provide an algorithm to compute it. We improve the algorithm in the case of braid groups, and discuss its complexity in this case. As a necessary result for obtaining the general algorithm, we prove that the centralizers of positive powers of an element form a periodic sequence and we show how to compute its period.

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