Towards Monoidal Categorifications of Twisted Products of Flag Varieties

TL;DR

This paper constructs a monoidal category of quantum affine algebra representations that encapsulates cluster algebra structures related to twisted products of flag varieties, including braid varieties and double Bruhat cells.

Contribution

It introduces a new monoidal categorification framework for cluster algebras associated with twisted flag variety products in quantum algebra.

Findings

Grothendieck ring contains a cluster algebra structure

Includes braid varieties and double Bruhat cells

Provides a categorification linking quantum groups and algebraic geometry

Abstract

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.