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 $\beta$ and $v\le \delta (\beta )$, we study the cluster algebra associated with the twisted product of flag varieties ${\mathring{\mathcal{Z}}}_{v,\beta }$. 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 ${\mathcal{C}}_{v,\beta }$ 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 ${\mathcal{C}}_{v,\beta }$. The reverse inclusion, which would give a full monoidal categorification, is left as a conjecture.

60 pages. We used ChatGPT to improve the language and presentation of this paper and to assist in checking the logical consistency of its arguments