The twist for Richardson varieties

TL;DR

This paper introduces a new twist automorphism for open Richardson varieties in complex semisimple groups, preserving positivity and connecting two conjectural cluster structures, thus advancing understanding of their algebraic and geometric properties.

Contribution

It constructs a novel twist automorphism for Richardson varieties, generalizing previous maps and linking different cluster structure conjectures.

Findings

The twist automorphism preserves totally positive parts.

A Chamber Ansatz formula for the twist is established.

The map unifies existing twist constructions in the literature.

Abstract

We construct the twist automorphism of open Richardson varieties inside the flag variety of a complex semisimple algebraic group. We show that the twist map preserves totally positive parts, and prove a Chamber Ansatz formula for it. Our twist map generalizes the twist maps previously constructed by Berenstein-Fomin-Zelevinsky, Marsh-Scott, and Muller-Speyer. We use it to explain the relationship between the two conjectural cluster structures for Richardson varieties studied by Leclerc and by Ingermanson.