On the fundamental group of open Richardson varieties

TL;DR

This paper computes the fundamental group of open Richardson varieties in flag manifolds, confirming a mirror symmetry prediction and providing explicit equations for related divisors.

Contribution

It provides the first computation of the fundamental group for these varieties and verifies a key mirror symmetry prediction by Hori.

Findings

Fundamental group of open Richardson varieties is computed.

Verification of Hori's mirror symmetry prediction.

Explicit equations for the Knutson-Lam-Speyer anti-canonical divisor.

Abstract

We compute the fundamental group of an open Richardson variety in the manifold of complete flags that corresponds to a partial flag manifold. Rietsch showed that these log Calabi-Yau varieties underlie a Landau-Ginzburg mirror for the Langlands dual partial flag manifold, and our computation verifies a prediction of Hori for this mirror. It is log Calabi-Yau as it isomorphic to the complement of the Knutson-Lam-Speyer anti-canonical divisor for the partial flag manifold. We also determine explicit defining equations for this divisor.