ON THE COMBINATORICS OF CRYSTAL GRAPHS, II. THE CRYSTAL COMMUTOR

We present an explicit combinatorial realization of the commutor in the category of crystals which was first studied by Henriques and Kamnitzer. Our realization is based on certain local moves defined by van Leeuwen. We work in the category g-Crystals of crystals corresponding to representations of complex semisimple Lie algebras g. It is well-known that this is a monoidal category with an associative tensor product. The crystals A ⊗B and B ⊗A are isomorphic via maps called commutors. The map flip : A ⊗B →B ⊗A , (a, b) ↦→(b, a) is not a commutor. Henriques and Kamnitzer, based on an idea of Bernstein, defined a commutor σA,B based on Lusztig’s involution on a crystal. They also proved that the category g-Crystals with this commutor is a coboundary category. Our explicit realization of the commutor σA,B based on local moves is proved to hold, in particular, for all simple Lie algebras g with the exception of those of type E8, F4, and G2; moreover, it is conjectured to hold for all semisimple Lie algebras.