University of Minnesota Combinatorics Seminar: 2022

Date: 04/15/2022

Speaker: Dan Douglas

Tropical Fock-Goncharov Coordinates for $SL_3$-Webs on Surfaces
For a finite-type surface $S$, we study a preferred basis for the commutative algebra $C[R_{SL3}(S)]$ of regular functions on the $SL_3(C)$-character variety, introduced by Sikora-Westbury. These basis elements come from the trace functions associated to certain tri-valent graphs embedded in the surface $S$. 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.

Tropical Fock-Goncharov Coordinates for $SL_3$-Webs on Surfaces

For a finite-type surface $S$, we study a preferred basis for the commutative algebra $C[R_{SL3}(S)]$ of regular functions on the $SL_3(C)$-character variety, introduced by Sikora-Westbury. These basis elements come from the trace functions associated to certain tri-valent graphs embedded in the surface $S$. 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.