# Generalized Minors and Tensor Invariants

## TL;DR

This paper links generalized minors on double Bruhat cells to tensor invariants, enabling the construction and analysis of cluster structures on configuration spaces of flags through weight identities.

## Contribution

It provides formulas relating generalized minors to tensor invariants and demonstrates how to derive quivers for cluster structures from these invariants.

## Key findings

- Formulas for generalized minors as tensor invariants
- Verification of weight identities for cluster structures
- Method to compute tensor invariant weights from quivers

## Abstract

Berenstein, Fomin and Zelevinsky defined functions on double Bruhat cells which they called generalized minors. By relating certain double Bruhat cells to configuration spaces of flags, we give formulas for these generalized minors as tensor invariants. This allows us to verify certain weight identities. The weights of the tensor invariants can then be used to construct the quiver for the cluster structure on the configuration space of three flags. We also show a converse statement--that the weights of tensor invariants can by computed from the structure of the quiver. The weight identities are important because they are necessary for the existence of cluster structures on the moduli space of framed local systems.