Combinatorial descriptions of the crystal structure on certain PBW bases

Ben Salisbury (1) , Adam Schultze (2) , Peter Tingley (3)

1 Department of Mathematics - University of Michigan
2 Department of Mathematics [New York CUNY]
3 Department of Mathematics and Statistics [Chicago]

Lusztig's theory of PBW bases gives a way to realize the crystal B(∞) for any simple complex Lie algebra where the underlying set consists of Kostant partitions. In fact, there are many different such realizations, one for each reduced expression for the longest element of the Weyl group. There is an algorithm to calculate the actions of the crystal operators, but it can be quite complicated. For ADE types, we give conditions on the reduced expression which ensure that the corresponding crystal operators are given by simple combinatorial bracketing rules. We then give at least one reduced expression satisfying our conditions in every type except E8, and discuss the resulting combinatorics. Finally, we describe the relationship with more standard tableaux combinatorics in types A and D.

Mots clés: Combinatorics

Domaines: Combinatoire [math.CO]

Ben Salisbury, Adam Schultze, Peter Tingley. Combinatorial descriptions of the crystal structure on certain PBW bases. 28-th International Conference on Formal Power Series and Algebraic Combinatorics, Simon Fraser University, Jul 2016, Vancouver, Canada. ⟨10.46298/dmtcs.6377⟩. ⟨hal-02166334⟩