Weyl group symmetry of q-characters

• Edward Frenkel ^[1] ; David Hernandez ^[2]
  1. [1] University of California System

     University of California System

     Estados Unidos

     
  2. [2] Université Paris Cité and Sorbonne Université, Paris, France
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 4, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-025-01072-3
• Enlaces
  • Texto completo
• Resumen

  • We define an action of the Weyl group WWW of a simple Lie algebra g\mathfrak{g}g on a completion of the ring Y\mathcal{Y}Y, which is the codomain of the qqq-character homomorphism of the corresponding quantum affine algebra Uq(g^)U_q(\widehat{\mathfrak{g}})Uq(g). We prove that the subring of WWW-invariants of Y\mathcal{Y}Y is precisely the ring of qqq-characters, which is isomorphic to the Grothendieck ring of the category of finite-dimensional representations of Uq(g^)U_q(\widehat{\mathfrak{g}})Uq(g). This resolves an old puzzle in the theory of qqq-characters. We also identify the screening operators, which were previously used to describe the ring of qqq-characters, as the subleading terms of simple reflections from WWW in a certain limit. Our results have already found applications to the study of the category O\mathcal{O}O of representations of the Borel subalgebra of Uq(g^)U_q(\widehat{\mathfrak{g}})Uq(g) in [Frenkel and Hernandez, Extended Baxter Relations and QQ-Systems for Quantum Affine Algebras, Comm. Math. Phys. 405:190, 2024. (arXiv:2312.13256)] and to the categorification of cluster algebras in [Geiss et al., Representations of shifted quantum affine algebras and cluster algebras I. The simply-laced case, Proc. Lond. Math. Soc. 3(129): e12630, 2024 (arXiv:2401.04616)].

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