Cluster algebras and Grassmannians of type G2

TL;DR

This paper proves a conjecture linking cluster algebra structures to the coordinate rings of certain algebraic varieties associated with the G2 Lie group, enhancing understanding of their algebraic and geometric properties.

Contribution

It establishes cluster algebra structures on the multi-homogeneous coordinate rings of partial flag varieties for G2, confirming a conjecture and refining known results about double Bruhat cells.

Findings

Proved that coordinate rings of G2 partial flag varieties have cluster algebra structures.

Confirmed that the coordinate ring of the double Bruhat cell G^{e,w_0} is a cluster algebra.

Produced explicit cluster algebra structures for G2-related varieties.

Abstract

We prove a conjecture of Geiss, Leclerc and Schr\"{o}er, producing cluster algebra structures on multi-homogeneous coordinate ring of partial flag varieties, for the case $G_2$. As a consequence we sharpen the known fact that coordinate ring of the double Bruhat cell $G^{e,w_0}$ is an upper cluster algebra, by proving that it is a cluster algebra.