Some results on the higher-rank graphs associated to crystals of semisimple Lie algebras

TL;DR

The paper investigates higher-rank graphs $\Gamma_{\mathfrak{g}}$ built from crystals of semisimple Lie algebras and clarifies how Bruhat graphs from the Weyl group embed into these graphs. It proves a unique embedding of the right weak Bruhat graph into the skeleton and a Bruhat embedding with compatible coloring into the full graph, with type $A$ revealing a tight link between right ends and Lascoux–Schützenberger keys via tableaux and jeu de taquin. In type $A$, right ends coincide with left LS-keys, and the Cartan braiding translates to combinatorial moves on skew tableaux, tying representation-theoretic structure to classical tableau theory. The results connect $C^*$-algebraic perspectives on quantum groups with combinatorial models, offering a framework to study higher-rank graph encodings of Lie-theoretic data and suggesting avenues for generalized keys beyond type $A$.

Abstract

In this paper we continue the study of the higher-rank graphs associated to finite-dimensional complex semisimple Lie algebras, introduced by the author and R. Yuncken, whose construction relies on Kashiwara's theory of crystals. First we prove that the Bruhat graphs of the corresponding Weyl groups, both weak and strong, can be embedded into the higher-rank graphs as colored graphs. Next, specializing to Lie algebras of type $A$, we connect some aspects of the construction of the higher-rank graphs with some well-known notions in combinatorics, most notably the keys of Lascoux and Schützenberger.

Figures (8)

• Figure 5.1: The left and right weak Bruhat graphs for the Weyl group $S_3$, considered as colored graphs. Here the red solid edges correspond to the color $1$, while the blue dashed edges correspond to the color $2$.
• Figure 5.2: The (strong) Bruhat graph $B(W)$ for $W = S_3$.
• Figure 5.3: The skeleton of the $2$-graph $\Gamma_{\mathfrak{g}}$ for $\mathfrak{g} = A_2$ (with the loops on each vertex removed for readability). The red solid edges correspond to the color $\omega_1$, while the blue dashed edges correspond to the color $\omega_2$.
• Figure 6.1: Jeu de taquin slides for the Cartan component of $\mathcal{B}(\omega_1) \otimes \mathcal{B}(\omega_2)$.
• Figure 6.2: The right ends of the various elements of $\mathcal{B}(\rho)$ for $\mathfrak{g} = A_2$.

Theorems & Definitions (61)

• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 3.1
• Example 3.2
• Definition 3.3
• Definition 3.4
• Remark 3.5
• Lemma 3.6
• Definition 3.7