Fock-Goncharov conjecture and polyhedral cones for $U\subset S{L}_n$ and base affine space $S{L}_n/U$

Abstract

I prove several conjectures of \cite{GHKK} on the cluster structure of $S{L}_n$, which in particular imply the full Fock-Goncharov conjecture for the open double Bruhat cell $\mathcal{A}\subset S{L}_n/U$, for $U\subset S{L}_n$ a maximal unipotent subgroup. This endows the mirror cluster variety $\mathcal{X}$ with a canonical potential function $W$, and determines a canonical cone $W^T\ge 0\subset \mathcal{X}\left({\mathbb{R}}^T\right)$ of the mirror tropical space, whose integer points parametrize a basis of $H^0\left(S{L}_n/U,{\mathcal{O}}_{S{L}_n/U}\right)$, canonically determined by the open subset $\mathcal{A}\subset S{L}_n/U$. Each choice of seed identifies $\mathcal{X}\left({\mathbb{R}}^T\right)$ with a real vector space, and $W^T\ge 0$ with a system of linear equations with integer coefficients, cutting out a polyhedral cone. We obtain in this way (generally) infinitely many parameterizations of the canonical basis as integer points of a polyhedral cone. For the usual initial seed of the double Bruhat cell, we recover the parametrizations of Berenstein-Kazhdan\cite{BKaz,BKaz2} and Berenstein-Zelevinsky\cite{BZ96} by integer points of the Gelfand-Tsetlin cone.

References in corpus (3)

• Cluster algebras IV: Coefficients
• Cluster algebras III: Upper bounds and double Bruhat cells
• Vertices of Gelfand-Tsetlin Polytopes

Cited by in corpus (1)

• Extending Upper Cluster Algebras