Cluster realization of positive representations of split real quantum Borel subalgebra

In our previous work, we studied the positive representations of split real quantum groups $\mathcal{U}_{q\tilde{q}}(\mathfrak{g}_\mathbb{R})$ restricted to its Borel part, and showed that they are closed under taking tensor products. However, the tensor product decomposition was only constructed abstractly using the GNS-representation of a $C^*$-algebraic version of the Drinfeld-Jimbo quantum groups. In this paper, using the recently discovered cluster realization of quantum groups, we write down the decomposition explicitly by realizing it as a sequence of cluster mutations in the corresponding quiver diagram representing the tensor product.