Braid group actions on grassmannians and extended crystals of type $A$

Abstract

Let $σ_i$ be the braid actions on infinite Grassmannian cluster algebras induced from Fraser's braid group actions. Let $\mathsf{T}_i$ be the braid group actions on (quantum) Grothendieck rings of Hernandez-Leclerc category ${\mathscr C}_\mathfrak{g}^0$ of affine type $A_n^{(1)}$, and $\mathsf{R}_i$ the braid group actions on the corresponding extended crystals. In the paper, we prove that the actions $σ_i$ coincide with the braid group actions $\mathsf{T}_i$ and $\mathsf{R}_i$.

Figures (1)

• Figure 1: An initial seed for $\mathbb{G}_4^{+}$. The $(i,j)$ on vertices are positions of the vertices.

Theorems & Definitions (40)

• Definition 2.1
• Remark 2.2
• Remark 2.3
• Definition 2.4
• Example 2.5
• Theorem 2.6: P23
• Example 2.7
• Remark 3.1
• Example 3.2
• Example 3.3