In this talk, we introduce a new class of toric Richardson varieties, which we call unexpected toric Richardson varieties. These Richardson varieties are toric varieties with respect to the action of a torus of higher dimension (compared to the standard torus case), whose torus action was previously completely unexplored. The standard torus action on Richardson varieties and a characterization of (expected) toric Richardsons with respect to the standard torus were well understood. The existence of such unexpected toric Richardsons were implied by our characterization result for all toric Richardson varieties: the open Richardson variety $R_{v,w}$ in the complete flag variety is a torus $\mathbb{T}$ if and only if the closed Richardson variety $\overline{R}_{v,w}$ is a toric variety with respect to $\mathbb{T}$. Equivalently, this can be said that when the Bruhat interval $[ v , w ] $ is a lattice. Moreover, we prove that similar to the standard torus case, the moment polytopes for all toric Richardson varieties have a nice combinatorial description in terms of Bruhat interval. This is based on my joint work with Eugene Gorsky and Melissa Sherman-Bennett.
