Quantum affine algebras and Grassmannians
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Mathematische Zeitschrift
Quantum affine algebras and Grassmannians Wen Chang1 · Bing Duan2 · Chris Fraser3 · Jian-Rong Li4,5 Received: 27 August 2019 / Accepted: 15 January 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We study the relation between quantum affine algebras of type A and Grassmannian cluster algebras. Hernandez and Leclerc described an isomorphism from the Grothendieck n )-modules to a quotient of the Grassmanring of a certain subcategory C of Uq (sl nian 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 + + 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 n )-modules. We give a tableau-theoretic rule for performfor finite-dimensional Uq (sl ing 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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Jian-Rong Li [email protected] Wen Chang [email protected] Bing Duan [email protected] Chris Fraser [email protected]
1
School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, China
2
School of Mathematics and Statistics, Lanzhou University, Lanzhou, China
3
School of Mathematics, University of Minnesota, Minneapolis, USA
4
Department of Mathematics and Scientific Computing, University of Graz, 8010 Graz, Austria
5
Department of Mathematics, The Weizmann Institute of Science, 7610001 Rehovot, Israel
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W. Chang et al.
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . 2 Hernandez–Leclerc’s category and the Grassmannian . 2.1 Cluster algebras . . . . . . . . . . . . . . . . . . 2.2 Quantum affine algebras . . . . . . . . . . . . .
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A
2.3 Finite-dimensional modules and the category C n−1 . . . . . . . . . . . . . . . . . . . . . . . . . A 2.4 Cluster structure on K 0 (C n−1 )
. . . . . . . . . . 2.5 Grassmannian cluster algbras . . . . . . . . . . . . 3 Simple Uq (ˆg)-modules and tableaux . . . . . . . . . . 3.1 Weight and partial order for tableaux . . . . . . . . 3.2 The tableau monoid . . . . . . . . . . . . . . . . . 3.3 Tableaux and modules . . . . . . . . . . . . . . . . 3.4 The elements ch(T )
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