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CAGE: Philadelphia Area Combinatorics and Alg. Geometry Seminar

Tuesday, May 10, 2016 - 3:00pm

Ben Salisbury

Central Michigan University

Location

Drexel University

Korman 245

Lusztig's theory of PBW bases gives a way to realize the crystal $B(\infty)$ for any complex-simple Lie algebra where the underlying set consists of Kostant partitions. In fact, there are many different such realizations: one for each reduced expression of the longest element of the Weyl group. There is an explicit algorithm to calculate the actions of the crystal operators, but it can be quite complicated. For simply-laced types, we give conditions on the reduced expression which ensure that the corresponding crystal operators are given by a simple combinatorial bracketing rule. We then give at least one reduced expression satisfying our conditions in every simply-laced type except $E_8$, and discuss the resulting combinatorics. Finally, we describe the relationship with more standard tableaux combinatorics in type $D$. This is joint work with Peter Tingley and Adam Schultze.