Crystal basis

A crystal basis is the combinatorial limit of a basis of a quantum-group representation. Its colored directed graph records the actions of the Kashiwara lowering operators.

On a crystal, ${\tilde{f}}_i$ follows an edge of color $i$ and ${\tilde{e}}_i$ follows that edge backwards.

The tensor product crystal uses the $i$-string data ${\epsilon }_i$ and ${\phi }_i$ to decide on which tensor factor a Kashiwara operator acts. Its connected components are highest-weight crystals and encode the decomposition of the tensor product representation.