Extension of cluster-induced torus actions from open to closed Richardson varieties

Determine whether, for every Bruhat interval [v,w] in S_n, the torus T arising from the cluster algebra structure on the open Richardson variety R_{v,w}—whose dimension equals the number of frozen variables—acts on the closed Richardson variety \overline{R}_{v,w} by extending its action from R_{v,w}.

Background

Open Richardson varieties R_{v,w} carry cluster algebra structures, and from these structures one can construct an action of a (potentially larger) algebraic torus T on R_{v,w}, where the dimension of T equals the number of frozen variables in the cluster. This motivates asking whether such a torus action extends from the open affine piece to its Zariski closure.

The paper proves that when R_{v,w} itself is a torus (the maximal-dimension case), the T-action extends to \overline{R}_{v,w}, yielding a toric Richardson variety. Equivalently, this occurs precisely when the Bruhat interval [v,w] is a lattice (i.e., has no 2-crown subinterval). Beyond this maximal case, the general extension question is not resolved by the results presented.