Tropical Fock-Goncharov coordinates for ${\mathrm{SL}}_3$-webs on surfaces I: construction

Abstract

For a finite-type surface $\mathfrak{S}$, we study a preferred basis for the commutative algebra $\mathbb{C}[{\mathcal{R}}_{{\mathrm{SL}}_3(\mathbb{C})}(\mathfrak{S})]$ of regular functions on the ${\mathrm{SL}}_3(\mathbb{C})$-character variety, introduced by Sikora-Westbury. These basis elements come from the trace functions associated to certain tri-valent graphs embedded in the surface $\mathfrak{S}$. We show that this basis can be naturally indexed by non-negative integer coordinates, defined by Knutson-Tao rhombus inequalities and modulo 3 congruence conditions. These coordinates are related, by the geometric theory of Fock and Goncharov, to the tropical points at infinity of the dual version of the character variety.

58 pages, 58 figures. Version 3: Final version after publication

References in corpus (1)

• Triangular decomposition of $S{L}_3$ skein algebras

Cited by in corpus (1)

• Naturality of ${\mathrm{SL}}_3$ quantum trace maps for surfaces