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

TL;DR

This paper explores simplified combinatorial rules for crystal operators in Lusztig's PBW bases for simple Lie algebras, providing conditions for easier computation and connections to standard tableaux in types A and D.

Contribution

It introduces conditions on reduced expressions that simplify crystal operator calculations and relates these to known combinatorial models.

Findings

Identifies conditions ensuring simple bracketing rules for crystal operators

Provides explicit reduced expressions satisfying these conditions for most types

Connects PBW bases combinatorics with standard tableaux in types A and D

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 $E_8$, and discuss the resulting combinatorics. Finally, we describe the relationship with more standard tableaux combinatorics in types A and D.