A cluster structure on the coordinate ring of partial flag varieties

TL;DR

This paper demonstrates that the coordinate rings of partial flag varieties for simply-connected semisimple complex groups can be endowed with a cluster algebra structure, extending previous results on Schubert cells.

Contribution

It constructs an explicit cluster algebra structure on the coordinate rings of partial flag varieties, generalizing known structures on Schubert cells.

Findings

Cluster algebra structure on coordinate rings of partial flag varieties.

Explicit lifting of cluster structures from Schubert cells to flag varieties.

Proof that the constructed cluster algebra equals the coordinate ring.

Abstract

The main goal of this paper is to show that the (multi-homogeneous) coordinate ring of a partial flag variety $\mathbb{C} [G / P_K^{-}]$ admits a cluster algebra structure if $G$ is any simply-connected semisimple complex algebraic group. We use derivation properties and a special lifting map to prove that the cluster algebra structure $\mathcal{A}$ of the coordinate ring $\mathbb{C}[N_K]$ of a Schubert cell constructed by Goodearl and Yakimov can be lifted, in an explicit way, to a cluster structure $\hat{\mathcal{A}}$ living in the coordinate ring of the corresponding partial flag variety. Then we use a minimality condition to prove that the cluster algebra $\hat{\mathcal{A}}$ is indeed equal to $\mathbb{C}[G / P_K^{-}]$.