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Higher order Kirillov--Reshetikhin modules for 𝐔 q (A n (1)), imaginary modules and monoidal categorification

  • Matheus Brito ORCID logo EMAIL logo and Vyjayanthi Chari
Published/Copyright: October 27, 2023
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Journal fĂźr die reine und angewandte Mathematik (Crelles Journal)
From the journal Journal fĂźr die reine und angewandte Mathematik (Crelles Journal) Volume 2023 Issue 804

Abstract

We study the family of irreducible modules for quantum affine 𝔰 ⁢ 𝔩 n + 1 whose Drinfeld polynomials are supported on just one node of the Dynkin diagram. We identify all the prime modules in this family and prove a unique factorization theorem. The Drinfeld polynomials of the prime modules encode information coming from the points of reducibility of tensor products of the fundamental modules associated to A m with m ≤ n . These prime modules are a special class of the snake modules studied by Mukhin and Young. We relate our modules to the work of Hernandez and Leclerc and define generalizations of the category 𝒞 - . This leads naturally to the notion of an inflation of the corresponding Grothendieck ring. In the last section we show that the tensor product of a (higher order) Kirillov–Reshetikhin module with its dual always contains an imaginary module in its Jordan–Hölder series and give an explicit formula for its Drinfeld polynomial. Together with the results of [D. Hernandez and B. Leclerc, A cluster algebra approach to q-characters of Kirillov–Reshetikhin modules, J. Eur. Math. Soc. (JEMS) 18 2016, 5, 1113–1159] this gives examples of a product of cluster variables which are not in the span of cluster monomials. We also discuss the connection of our work with the examples arising from the work of [E. Lapid and A. Mínguez, Geometric conditions for □ -irreducibility of certain representations of the general linear group over a non-archimedean local field, Adv. Math. 339 2018, 113–190]. Finally, we use our methods to give a family of imaginary modules in type D 4 which do not arise from an embedding of A r with r ≤ 3 in D 4 .

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1719357

Funding statement: Vyjayanthi Chari was partially supported by the National Science Foundation (DMS-1719357), the Max Planck Institute, Bonn and by the InfoSys Visiting Chair position at the Indian Institute of Science. Both authors were supported by the Oberwolfach Research Fellows program and acknowledge the superb working conditions at the Mathematisches Forschungsinstitut Oberwolfach.

Acknowledgements

The authors thank David Hernandez and Bernard Leclerc for many interesting discussions on their work over the years. The authors also thank the referee for their careful reading of the paper and the many helpful comments and remarks. In particular, the referee has pointed out that there might be analogous statements for the generalizations in [21] of the category 𝒞 - and we hope to return to this in future work. Matheus Brito thanks the Hausdorff Research Institute for Mathematics and the organizers of the Trimester Program “New Trends in Representation Theory” for excellent working conditions.

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Received: 2022-12-05
Revised: 2023-07-29
Published Online: 2023-10-27
Published in Print: 2023-11-01

Š 2023 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Proof of the Michael–Simon–Sobolev inequality using optimal transport
  3. Existence and symmetry of periodic nonlocal-CMC surfaces via variational methods
  4. Semi-continuity of conductors, and ramification bound of nearby cycles
  5. The integrality conjecture and the cohomology of preprojective stacks
  6. Graphical solutions to one-phase free boundary problems
  7. Quasi-projectivity of images of mixed period maps
  8. Higher order Kirillov--Reshetikhin modules for 𝐔 q (A n (1)), imaginary modules and monoidal categorification
  9. The set of local 𝐴-packets containing a given representation
https://doi.org/10.1515/crelle-2023-0068

Articles in the same Issue

  1. Frontmatter
  2. Proof of the Michael–Simon–Sobolev inequality using optimal transport
  3. Existence and symmetry of periodic nonlocal-CMC surfaces via variational methods
  4. Semi-continuity of conductors, and ramification bound of nearby cycles
  5. The integrality conjecture and the cohomology of preprojective stacks
  6. Graphical solutions to one-phase free boundary problems
  7. Quasi-projectivity of images of mixed period maps
  8. Higher order Kirillov--Reshetikhin modules for 𝐔 q (A n (1)), imaginary modules and monoidal categorification
  9. The set of local 𝐴-packets containing a given representation
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