nLab
crystal basis

Crystal bases are a construction in the representation theory of quantum groups (which in a specialization exist hence for usual Lie groups as well): roughly speaking they provide a uniform description not only of irreducible finite-dimensional modules but also a uniform choice of bases in all of them as well as in tensor products.

There are two mutually dual versions due Masaki Kashiwara and George Lusztig.

History and references

The terminology is due to the involved q=0q=0 limit of quantum groups used in the construction (the classical case is q=1q=1). In a thermodynamic parlance zero temperature would involve passing to crystalization.