Towards Monoidal Categorifications of Twisted Products of Flag Varieties

Let $G$ be a simple, simply connected, simply laced algebraic group. We construct a monoidal category of representations of the quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ whose Grothendieck ring contains a cluster algebra with initial seed given by that of the coordinate ring of twisted products of flag varieties. This class of varieties includes, in particular, braid varieties and reduced double Bruhat cells.