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

Cluster realization of positive representations of split real quantum Borel subalgebra