Cluster Decorated Geometric Crystals, Generalized Geometric RSK-Correspondences, and Donaldson-Thomas Transformations

  • Gleb Koshevoy
Part of the MATRIX Book Series book series (MXBS, volume 2)


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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I thank Arkady Berenstein, Volker Genz and Bea Schumann for inspired and fruitful discussions, organizers of the MATRIX workshop, and especially Paul Zinn-Justin, and the RSF grant 16-11-10075 for financial support.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Gleb Koshevoy
    • 1
  1. 1.IITP RASMCCME, and Interdisciplinary Scientific Center J.-V. PonceletMoscowRussia

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