GLS homogenization tilde map

TL;DR

This paper investigates the GLS homogenization tilde map in cluster algebras, providing an explicit algorithm for calculating it on generalized minors, thereby enhancing understanding of its role in lifting cluster structures.

Contribution

It offers a detailed analysis and an explicit computational algorithm for the GLS tilde map on generalized minors within cluster algebras.

Findings

Explicit algorithm for GLS tilde map on generalized minors

Enhanced understanding of the tilde map's role in cluster algebra lifting

Practical method for calculations in partial flag varieties

Abstract

In the construction of a cluster algebra on the homogeneous coordinate ring of a partial flag variety by Gei{\ss}, Leclerc and Schr{\"{o}}er, they defined a special map denoted by ``tilde". This map lifts each element $f$ of the coordinate ring of a Schubert cell uniquely to an element $\widetilde{f}$ of the (multi-homogeneous) coordinate ring of the corresponding partial flag variety. The significance of this map appears from its essential role; it lifts the cluster algebra of the coordinate ring of a cell to a cluster algebra living in the coordinate ring of the corresponding partial flag variety. This paper takes a closer look at this map and gives an explicit algorithm to calculate it for the \textit{generalized minors}.