Generalized Minors and Tensor Invariants

Abstract

Abstract In the paper [1], the authors define functions on double Bruhat cells, which they call generalized minors. By relating certain double Bruhat cells to the spaces $\operatorname {Conf}_3 {\mathcal {A}}_G$ and $\operatorname {Conf}_4 {\mathcal {A}}_G$, we give formulas for these generalized minors as tensor invariants. This allows us to verify certain weight identities conjectured in [7]. We then show that the grading of these tensor invariants is equivalent information to the quiver for the cluster structure on $\operatorname {Conf}_3 {\mathcal {A}}_G$. This leads to a simple combinatorial construction of cluster structures on the moduli space of framed local systems ${\mathcal {X}}_{G^{\prime},S}$ and ${\mathcal {A}}_{G,S}$.