Combinatorial descriptions of the crystal structure on certain PBW bases

TL;DR

This work develops a unified combinatorial framework to realize the crystal $B(\infty)$ via PBW bases and Kostant partitions for semisimple Lie algebras. By introducing simply braided reduced expressions, the authors show that crystal operators can be computed with simple bracketing rules derived from rank-2 subroot systems, reducing complex braid-move manipulations to bracket manipulations. They prove existence of simply braided words in all types except possibly $E_8$, $F_4$, and $G_2$, and provide detailed bracketing rules for rank-2 cases and for classical types $A_n$, $D_n$, $B_n$, and $C_n$, including explicit examples. The results offer a practical, uniform method to compute crystal operations on Kostant partitions, with potential broader implications for quiver perspectives and canonical basis realizations across Lie 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.

Figures (2)

• Figure 2.1: Dynkin diagrams of classical type following Bourbaki bourbaki.
• Figure 4.1: The inductive step in the proof of Theorem \ref{['th:brgeneral']}. Here we are applying $f_3$ in type $A_3$. At each step we have placed the string of brackets $S_3( {\bm c} )$ below the monomial $F^{\bm c}$ (note, for example, that the brackets for the roots $F_{12}$ and $F_{123}$ need to be reversed). This is the final case, where the first uncanceled bracket is in ${\bm c}^1$. The first braid move is the move $M_1$, and involves the roots $2$, $23$, and $3$. The new string of brackets after that move is obtained by deleting the brackets corresponding to these three roots, and replacing them with the number of uncanceled right brackets in the string we would use to apply $f_3$ in that rank 2 case. There were no uncanceled left brackets further to the left, so now there are no uncanceled left brackets at all, and, by induction, $f_3$ increases the exponent of $F_3$ by $1$. Then we undo the move $M_1$. The result is the result of applying $f_3$ to the rank 2 monomial $F_2^{(2)} F_{23}^{(3)} F_3^{(1)}$.

Theorems & Definitions (57)

• Proposition 2.2
• Corollary 2.3: TW12
• Definition 2.4
• Theorem 2.5: papi
• Lemma 2.6
• proof
• Definition 2.7
• Example 2.8
• Lemma 2.9
• proof