Abstract

For a simply connected, connected, semisimple complex algebraic group G, we define two geometric crystals on the \(\mathscr A\)-cluster variety of double Bruhat cell B−∩ Bw0B. These crystals are related by the ∗ duality. We define the graded Donaldson-Thomas correspondence as the crystal bijection between these crystals. We show that this correspondence is equal to the composition of the cluster chamber Ansatz, the inverse generalized geometric RSK-correspondence, and transposed twist map due to Berenstein and Zelevinsky.

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