Tropical Fock-Goncharov coordinates for SL_3-webs on surfaces

Abstract: For a finite-type surface $\mathfrak{S}$, we study a preferred basis for the commutative algebra $\mathbb{C}[\mathcal{R}_{\mathrm{SL}_3}(\mathfrak{S})]$ of regular functions on the $\text{SL}_3(\mathbb C)$-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 positive integer coordinates, defined by Knutson-Tao rhombus inequalities and modulo $3$ congruence conditions. These coordinates are related, by the geometric theory of Fock-Goncharov, to the tropical points at infinity of the dual version of the character variety. This is joint work with Zhe Sun.

mathematical physicscommutative algebraalgebraic geometrycombinatoricsquantum algebrarepresentation theorysymplectic geometry

Audience: researchers in the topic

PIMS Geometry / Algebra / Physics (GAP) Seminar