# Coboundary categories and local rules

## TL;DR

This paper develops the theory of local rules for coboundary categories, describing their application to quantum groups, Hecke algebra representations, and crystals, and extends growth diagrams to the cactus group action.

## Contribution

It introduces a comprehensive framework for local rules in coboundary categories and applies it to quantum groups, Hecke algebras, and crystals, including minuscule cases.

## Key findings

- Local rules for coboundary categories are formulated.
- Application of local rules to quantum groups and Hecke algebra representations.
- Extension of growth diagrams to cactus group actions on highest weight words.

## Abstract

First we develop the theory of local rules for coboundary categories. Then we describe the local rules in two main cases. First for the quantum groups in general and in the seminormal representations of the Hecke algebras. Then for crystals in general and specifically for crystals of minuscule representations. Finally we show how growth diagrams can be extended to construct the action of the cactus group on highest weight words.