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.
Salisbury, Ben; Schultze, Adam; and Tingley, Peter. Combinatorial Descriptions of the Crystal Structure on Certain PBW Bases (Extended Abstract). Proceedings for FPSAC 2016, , : 1-13, 2016. Retrieved from Loyola eCommons, Mathematics and Statistics: Faculty Publications and Other Works,
Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License.
© The Authors 2016