# On the wildness of cambrian lattices

## TL;DR

This paper classifies the representation types of cambrian lattices based on the Coxeter group's rank and structure, revealing a simple trichotomy from finite to wild types.

## Contribution

It provides a clear classification of cambrian lattices' representation types, highlighting the conditions under which they are finite, tame, or wild.

## Key findings

- Finite type when rank ≤ 2
- Tame type for reducible type A3 lattices
- Wild type in all other cases

## Abstract

In this note, we investigate the representation type of the cambrian lattices and some other related lattices. The result is expressed as a very simple trichotomy. When the rank of the underlined Coxeter group is at most 2, the lattices are of finite representation type. When the Coxeter group is a reducible group of type A 3 1 , the lattices are of tame representation type. In all the other cases they are of wild representation type.