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Lusztig's theory of PBW bases gives a way to realize the infinity crystal forany simple complex Lie algebra where the underlying set consists of Kostantpartitions. In fact, there are many different such realizations, one for eachreduced expression for the longest element of the Weyl group. There is analgorithm to calculate the actions of the crystal operators, but it can bequite complicated. For ADE types, we give conditions on the reduced expressionwhich ensure that the corresponding crystal operators are given by simplecombinatorial bracketing rules. We then give at least one reduced expressionsatisfying our conditions in every type except $E_8$, and discuss the resultingcombinatorics. Finally, we describe the relationship with more standardtableaux combinatorics in types A and D.

Combinatorial descriptions of the crystal structure on certain PBW bases (extended abstract)