Local acyclicity of Richardson-variety seeds in Kac–Moody type

Prove that, for every pair of Weyl-group elements v≤w and every reduced expression for w, the seed (v,\mathbf{w}) constructed for the open Richardson variety \mathring{\mathcal{B}}_{v,w} is locally acyclic, thereby establishing in general that its upper cluster algebra coincides with its cluster algebra.

Background

The paper constructs an upper cluster algebra structure on the coordinate ring of every open Richardson variety \mathring{\mathcal{B}}_{v,w} for a symmetrizable Kac–Moody group. The construction produces a seed (v,\mathbf{w}) from a reduced expression \mathbf{w} for w by iterated mutation, freezing, and deletion.

For finite-type groups, prior work established that the cluster structure on every open Richardson variety is locally acyclic. The authors conjecture that the same property holds for arbitrary Kac–Moody types. Local acyclicity is significant because it would imply equality between the upper cluster algebra and the cluster algebra associated with the seed. The paper verifies the equality in the special case w=vu with additive length, but leaves the general case unresolved.