The starting point for the ideas in this talk is that the Plactic monoid admits several descriptions, one of them in terms of, so-called, Kashiwara crystals. Kashiwara crystals are certain recursively-constructed edge-coloured directed graphs, built from a given crystal basis. The way in which crystals give rise to monoids can be described in purely combinatorial terms. In previous work, we modified this construction allowing us to characterize other monoids using this approach. In this talk, we will present these results and a more general idea of building monoids from crystals.
This is joint work with Alan Cain and Robert Gray.