A generalization of plactic-coplactic equivalences and Kazhdan-Lusztig cells

We propose a generalization of plactic (Knuth) and coplactic (dual-Knuth) equivalence relations to finitely generated Coxeter groups. We obtain a decomposition of the left cells by coplactic classes which agrees with two induction properties of the cells. In the case of simply laced Coxeter groups, we give a crochet procedure which is a useful tool to study the structure of plactic and coplactic classes, and helps construct bijections between them.