Combinatorial descriptions of the crystal structure on certain PBW bases

TL;DR

This paper simplifies the description of crystal operators in PBW bases for semisimple Lie algebras by introducing bracketing rules for specific reduced expressions, making calculations more accessible.

Contribution

It introduces a new bracketing rule for crystal operators applicable to certain reduced expressions, expanding the understanding of crystal structures in Lie algebra representations.

Findings

Bracketing rules simplify crystal operator calculations.

Existence of suitable reduced expressions in all types except possibly E8, F4, G2.

Examples illustrating the new rules across different types.

Abstract

Using the theory of PBW bases, one can realize the crystal $B(\infty)$ for any semisimple Lie algebra over $\mathbf{C}$ using Kostant partitions as the underlying set. In fact there are many 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. Here we show that, for certain reduced expressions, the crystal operators can also be described by a much simpler bracketing rule. We give conditions describing these reduced expressions, and show that there is at least one example in every type except possibly $E_8$, $F_4$ and $G_2$. We then discuss some examples.