Quantum unipotent subgroup and dual canonical basis

Kimura, Yoshiyuki

doi:10.1215/21562261-1550976

Cited by 74 publications

(91 citation statements)

References 55 publications

(111 reference statements)

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“…Kang, Kashiwara, Kim, and Oh [36] proved that the quantum unipotent coordinate algebra has a monoidal categorification as conjectured in [26,38]. The connection between monoidal categorification and quantum affine algebras is as follows.…”

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confidence: 84%

Quantum affine algebras and Grassmannians

et al. 2020

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We study the relation between quantum affine algebras of type A and Grassmannian cluster algebras. Hernandez and Leclerc described an isomorphism from the Grothendieck ring of a certain subcategory C ℓ of U q ( sl n )-modules to a quotient of the Grassmannian cluster algebra in which certain frozen variables are set to 1. We explain how this induces an isomorphism between the monoid of dominant monomials, used to parameterize simple modules, and a quotient of the monoid of rectangular semistandard Young tableaux with n rows and with entries in [n + ℓ + 1]. Via the isomorphism, we define an element ch(T ) in a Grassmannian cluster algebra for every rectangular tableau T . By results of Kashiwara, Kim, Oh, and Park, and also of Qin, every Grassmannian cluster monomial is of the form ch(T ) for some T . Using a formula of Arakawa-Suzuki, we give an explicit expression for ch(T ), and also give explicit q-character formulas for finite-dimensional U q ( sl n )-modules. We give a tableau-theoretic rule for performing mutations in Grassmannian cluster algebras. We suggest how our formulas might be used to study reality and primeness of modules, and compatibility of cluster variables.

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confidence: 84%

Quantum affine algebras and Grassmannians

et al. 2020

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“…It is known that any products of the elements D((i, r), 0), (i, r) ∈ I tw,♭ ξ belong to v Z/2 B up [45,Theorem 6.26]. Therefore, D and − →…”

Section: Corollaries Of the Isomorphism φmentioning

confidence: 99%

Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm

Hernandez

Oya

2019

Advances in Mathematics

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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

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“…It was recently proved in [9] that cluster monomials in A Q are always linearly independent over Z. Moreover, cluster monomials play a prominent role in the construction of Z-linear bases of A Q which are of interest with respect to the study of dual canonical bases of quantum groups; see, for instance, [20,22,24,26,27,33].…”

Section: Cluster Algebrasmentioning

confidence: 99%