| Summary: | For a finite-type surface
$\mathfrak {S}$
, we study a preferred basis for the commutative algebra
$\mathbb {C}[\mathscr {R}_{\mathrm {SL}_3(\mathbb {C})}(\mathfrak {S})]$
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 trivalent graphs embedded in the surface
$\mathfrak {S}$
. We show that this basis can be naturally indexed by nonnegative 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.
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