Abstract: Kuperberg introduced oriented 3-valent graphs on the surface, called 3-webs, to study the SL_3-invariant tensor products T of irreducible representations of SL_3. Then Kuntson-Tao found a family of linear inequalities to characterize when T contains an invariant vector. Let A be a variation of the SL_3 character variety which generalizes the Penner's decorated Teichmuller space. Actually, Goncharov--Shen related Kuntson-Tao inequalities to the positivity of A. On the surface, we identify the space of 3-webs up to homotopy with certain lattice of A mapping class group equivariantly. As a consequence, as predicted by Fock--Goncharov duality conjecture, these tropical points parameterize a linear basis of the regular function ring of the dual space explicitly. This is a joint work with Daniel Douglas.

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