Towards Monoidal Categorifications of Twisted Products of Flag Varieties

Let $G$ be a simple, simply connected algebraic group of simply-laced type. For a positive braid word $β$ and $v\leδ(β)$, we study the cluster algebra associated with the twisted product of flag varieties $\mathring{\mathcal Z}_{v,β}$. We compare its Bao--Ye seed with a right-inductive weave seed an