GHK mirror symmetry, the Knutson-Tao hive cone, and Littlewood-Richardson coefficients

Abstract

I prove that the full Fock-Goncharov conjecture holds for Conf₃[superscript x] ([mathcal] A)-- the configuration space of triples of decorated flags in generic position. As a key ingredient of this proof, I exhibit a maximal green sequence for the quiver of the initial seed. I compute the Landau-Ginzburg potential W on Conf₃[superscript x] ([mathcal] A)[superscript vee] associated to the partial minimal model Conf₃[superscript x] ([mathcal] A) [subset] Conf₃ ([mathcal] A). The integral points of the associated "cone" [Xi] [does not equal] {W[superscript T] [less than or equal to] 0] [subset] Conf₃[superscript x] ([mathcal] A)[superscript vee] ([mathbb R][superscript T]) parametrize a basis for [mathcal O] (Conf₃[superscript x] ([mathcal] A) )= [big o plus] (V[subscript alpha] [o times] V[supscript beta] [o times] V[subscript gamma])[subscript G] and encode the Littlewood-Richardson coefficients c[superscript gamma][subscript alpha beta]. I exhibit a unimodular p[superscript *] map that identifies W with the potential of Goncharov-Shen on Conf₃[superscript x] ([mathcal] A) and Xi with the Knutson-Tao hive cone.