Grassmannian cluster categories are studied in A categorification of Grassmannian cluster algebras and Cluster categories from Grassmannians and root combinatorics. The category $CM(B_{k,n})$ of Cohen-Macaulay modules over certain algebra $B_{k,n}$ is an additive categorification of the Grassmannian cluster algebra $\mathbb{C}[Gr(k,n)]$. A cluster character $M \mapsto \psi_M$ on the Grassmannian cluster category $CM(B_{k,n})$ is defined in Definition 9.1 A categorification of Grassmannian cluster algebras.
I think that the following multiplication formula is true:
For any Auslander-Reiten sequence $\tau M \to N \to M$ in the Auslander-Reiten sequence of $CM(B_{k,n})$, $$ \psi_{\tau M} \psi_M = \psi_N + \prod_i \psi_{P_i}, \quad (1) $$ where $P_i$ are some projective modules in $CM(B_{k,n})$ (they corresponds to frozen variables in $\mathbb{C}[Gr(k,n)]$). Is the formula (1) already known in literature? Thank you very much.
