A Newton-Okounkov body is a convex body constructed from a polarized variety with a valuation on its function field. Kaveh (resp., the first author and Naito) proved that the Newton-Okounkov body of a Schubert variety associated with a specific valuation is identical to the Littelmann string polytope (resp., the Nakashima-Zelevinsky polyhedral realization) of a Demazure crystal. These specific valuations are defined algebraically to be the highest term valuations with respect to certain local coordinate systems on a Bott-Samelson variety. Another class of valuations, which is geometrically natural, arises from some sequence of subvarieties of a polarized variety. In this paper, we show that the highest term valuation used by Kaveh (resp., by the first author and Naito) and the valuation coming from a sequence of specific subvarieties of the Schubert variety are identical on a perfect basis with some positivity properties. The existence of such a perfect basis follows from a categorification of the negative part of the quantized enveloping algebra. As a corollary, we prove that the associated Newton-Okounkov bodies coincide through an explicit affine transformation.Comment: 21 pages, to appear in J. London Math. Soc. (2

We establish ring isomorphisms between quantum Grothendieck rings of certain remarkable monoidal categories C Q,Bn and CQ,A 2n−1 of finite-dimensional representations of quantum affine algebras of types B 10 4. Quantum tori 19 5. Quantum tori for quantized coordinate algebras 23 6. The isomorphism between quantum tori 26 7. Quantum Grothendieck rings 31 8. Quantized coordinate algebras 35 9. Quantum T -system for type B 38 10. The isomorphism Φ 40 11. Corollaries of the isomorphism Φ 44 12. Comparison between type A and B 48 Appendix A. The inverse of the quantum Cartan matrix of type B n 56 Appendix B. Quantum cluster algebras 59 Appendix C. Unipotent quantum minors 60 References 62

In this paper, we construct twist automorphisms on quantum unipotent cells, which are quantum analogues of the Berenstein-Fomin-Zelevinsky twist automorphisms on unipotent cells. We show that those quantum twist automorphisms preserve the dual canonical bases of quantum unipotent cells.Moreover we prove that quantum twist automorphisms are described by the syzygy functors for representations of preprojective algebras in the symmetric case. This is the quantum analogue of Geiß-Leclerc-Schröer's description, and Geiß-Leclerc-Schröer's results are essential in our proof. As a consequence, we show that quantum twist automorphisms are compatible with quantum cluster monomials. The 6-periodicity of specific quantum twist automorphisms is also verified.

The theory of Newton-Okounkov bodies is a generalization of that of Newton polytopes for toric varieties, and it gives a systematic method of constructing toric degenerations of projective varieties. In this paper, we study Newton-Okounkov bodies of Schubert varieties from the theory of cluster algebras. We construct Newton-Okounkov bodies using specific valuations which generalize extended g-vectors in cluster theory, and discuss how these bodies are related to string polytopes and Nakashima-Zelevinsky polytopes.

For a symmetrizable Kac-Moody Lie algebra g, we construct a family of weighted quivers Q m (g) (m ≥ 2) whose cluster modular group Γ Qm(g) contains the Weyl group W (g) as a subgroup. We compute explicit formulae for the corresponding cluster Aand X -transformations. As a result, we obtain green sequences and the cluster Donaldson-Thomas transformation for Q m (g) in a systematic way when g is of finite type. Moreover if g is of classical finite type with the Coxeter number h, the quiver Q kh (g) (k ≥ 1) is mutation-equivalent to a quiver encoding the cluster structure of the higher Teichmüller space of a once-punctured disk with 2k marked points on the boundary, up to frozen vertices. This correspondence induces the action of direct products of Weyl groups on the higher Teichmüller space of a general marked surface. We finally prove that this action coincides with the one constructed in [GS16] from the geometrical viewpoint. Contents 1. Introduction 1 2. Notation and definitions in cluster algebra 5 3. Weyl group action 11 4. Quivers corresponding to reduced words 24 5. Application to the higher Teichmüller theory 30 6. Relation with the D g -quiver 52 Appendix A. Description of functions on Conf 3 A G 62 References 66