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Book chapter

Cluster decorated geometric crystals, generalized geometric RSK-correspondences, and Donaldson-Thomas transformations

P. 450-473.
Gleb Koshevoy.

For a simply connected, connected, semisimple complex algebraic group G,
we de ne two geometric crystals on the A-cluster variety of double Bruhat cell
B\cap \ Bw_0B. These crystals are related by the  duality . We de ne 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 cham-
ber Ansatz, the inverse generalized geometric RSK-correspondence, and transposed
twist map due to Berenstein and Zelevinsky.