A local characterization of crystal graphs

Crystal graphs are an especially nice way to encode the data one needs to compute tensor product multiplicities for (say) irreps of semisimple Lie groups or algebras. There are a number of known ways to generate crystal graphs (e.g., Littelmann paths, or various flavors of tableaux in the classical cases), but until now, there has not been a simple intrinsic description of these graphs.

In the talk, we will explain how to do this in the simply-laced case. The multiply-laced case remains open.

Here is a crystal graph of a 20-dimensional irrep of Spin(5).