Towards Monoidal Categorifications of Twisted Products of Flag Varieties

Abstract

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 and obtain local acyclicity and equality of the cluster and upper cluster algebras. Using Lusztig parameters in a bosonic extension algebra, we construct a monoidal subcategory $\mathscr C_{v,β}$ of a Hernandez--Leclerc category and prove that its Grothendieck ring contains the integral cluster algebra with noninvertible frozen variables. Every cluster monomial is the class of a real simple object of $\mathscr C_{v,β}$. The reverse inclusion, which would give a full monoidal categorification, is left as a conjecture.