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

Let ${\sigma }_i$ 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 ${\mathcal{C}}_{\mathfrak{g}}^0$ 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 ${\sigma }_i$ coincide with the braid group actions ${\mathsf{T}}_i$ and ${\mathsf{R}}_i$.