Document Type

Conference Proceeding

Publication Date


Publication Title

Proceedings for FPSAC 2016




Lusztig's theory of PBW bases gives a way to realize the infinity crystal 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.


Author Posting. © The Authors 2016. This article is posted here by permission of the authors for personal use, not for redistribution. The article was published in the Proceedings for FPSAC 2016,

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Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License.