Hurwitz Orbits with a Cellular Automaton

TL;DR

This paper introduces a cellular automaton approach to analyze Hurwitz orbits, enabling estimation of minimal plagues, which are key combinatorial invariants used in classifying Nichols algebras related to braid group actions.

Contribution

It presents a novel cellular automaton method for calculating plagues on Hurwitz orbits, advancing the combinatorial tools for Nichols algebra classification.

Findings

The cellular automaton effectively estimates minimal plagues.

The method improves understanding of braid group actions on racks.

It facilitates classification of Nichols algebras.

Abstract

Hurwitz orbits are the orbits of the braid group action on the powers of a rack. Hurwitz orbits for the action of the braid group on three strands are used in \cite{21} and \cite{22} for the classification of Nichols algebras. This classification is based on a combinatorial invariant called plague on the Hurwitz orbits. The method to calculate plagues on the Hurwitz orbits is formulated in \cite{22} by using a cellular automaton on the Hurwitz orbits and their quotients. By using this cellular automaton-based method we estimate the minimal plagues on the Hurwitz orbits.