Title

Combinatorial Descriptions of the Crystal Structure on Certain PBW Bases (Extended Abstract)

Authors

Ben Salisbury, Central Michigan University
Adam Schultze, University at Albany, State University of New York
Peter Tingley, Loyola University ChicagoFollow

Document Type

Conference Proceeding

Publication Date

3-31-2016

Publication Title

Proceedings for FPSAC 2016

Pages

1-13

Abstract

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.

Comments

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, https://arxiv.org/abs/1603.09013

Recommended Citation

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.

Copyright Statement

© The Authors 2016