Unexpected toric Richardson varieties

TL;DR

This paper characterizes when open Richardson varieties in complete flag varieties are torus-like, linking their structure to Bruhat interval combinatorics and providing explicit polytope descriptions.

Contribution

It establishes a criterion for open Richardson varieties to be toric, connecting their structure to Bruhat intervals and classifying the associated polytopes.

Findings

Open Richardson varieties are toric iff their closed counterparts are toric.

Classification of toric Richardson varieties via Bruhat interval combinatorics.

Explicit descriptions of the polytopes associated with these varieties.

Abstract

We prove that an open Richardson variety in the complete flag variety for $\mathrm{GL}_n$ is isomorphic to a torus if and only if the corresponding closed Richardson variety is toric. Such toric varieties can be classified in terms of the combinatorics of Bruhat intervals, and include many varieties of dimension larger than $n-1$. We give a combinatorial description of the corresponding polytopes, and compute several explicit examples.