Cluster structures on schemes of bands

We introduce new objects, called $(G,c)$-bands, associated with a simple simply-connected algebraic group $G$, and a Coxeter element $c$ in its Weyl group. We show that bands of a given type are the $K$-points of an infinite dimensional affine scheme, whose ring of regular functions has a cluster algebra structure. We also show that two important invariant sub-algebras of this ring are cluster sub-algebras. These three cluster structures have already appeared in different contexts related to the representation theories of quantum affine algebras, their Borel sub-algebras, and shifted quantum affine algebras. In this paper we show that they all belong to a common geometric setting.