On the combinatorics of crystal graphs, II. The crystal commutor

We present an explicit combinatorial realization of the commutor in the category of crystals, a concept initially examined by Henriques and Kamnitzer. Our objective is to construct this realization utilizing specific local moves delineated by van Leeuwen. The methodology involves working within the framework of g-Crystals, which are associated with representations of complex semisimple Lie algebras. Our work elaborates on the relationship between the standard maps known as commutors and the newly defined commutor σA,B based on Lusztig's involution, with an emphasis on its implications for coboundary categories. Significant results emerged, demonstrating that the proposed realization holds true for all simple Lie algebras except for those of type E8, F4, and G2, along with a conjecture suggesting its validity for all semisimple Lie algebras. This builds upon prior constructions reliant on intricate mappings of crystals, endorsing the hypothesis that the interplay between representations and combinatorial structures can yield new insights into the properties of Lie algebras.