In this paper, we analyze the transition maps for Lusztig's parameterization of the global basis for the bosonic extension $\widehat{ \mathcal{ A}}(b)$ of the quantum coordinate ring of a unipotent group, where $b$ belongs to the positive braid group $\operatorname{ Br}^+$. By examining different reduced expressions of $b$, we establish that the cluster structure on $\widehat{ \mathcal{ A}}(b)$ remains invariant regardless of the chosen reduced expression. For the simply-laced case, we construct a cluster structure on $\widehat{ \mathcal{ A}}(b)$ for all $b \in \operatorname{ Br}^+$, using a proof distinct from Qin's \cite{ qin2024based}.
Categories: math.RT