Skip to main content
SpringerLink
Log in

Cuspidal systems for affine Khovanov–Lauda–Rouquier algebras

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

A cuspidal system for an affine Khovanov–Lauda–Rouquier algebra \(R_\alpha \) yields a theory of standard modules. This allows us to classify the irreducible modules over \(R_\alpha \) up to the so-called imaginary modules. We describe minuscule imaginary modules, laying the groundwork for future study of imaginary Schur–Weyl duality. We introduce colored imaginary tensor spaces and reduce a classification of imaginary modules to one color. We study the characters of cuspidal modules. We show that under the Khovanov–Lauda–Rouquier categorification, cuspidal modules correspond to dual root vectors.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Baumann, P., Kamnitzer, J., Tingley, P.: Affine Mirković–Vilonen Polytopes. arXiv:1110.3661

  2. Beck, J.: Convex bases of PBW type for quantum affine algebras. Commun. Math. Phys. 165, 193–199 (1994)

    Article  MATH  Google Scholar 

  3. Beck, J., Chari, V., Pressley, A.: An characterization of the affine canonical basis. Duke Math. J. 99, 455–487 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Benkart, G., Kang, S.-J., Oh, S.-J, Park, E.: Construction of irreducible representations over Khovanov–Lauda–Rouquier algebras of finite classical type. Int. Math. Res. Not. (to appear). arXiv:1108.1048

  5. Brundan, J., Kleshchev, A.: Representation theory of symmetric groups and their double covers. In: Groups, Combinatorics and Geometry (Durham, 2001), pp. 31–53. World Scientific, River Edge (2003)

  6. Brundan, J., Kleshchev, A., McNamara, P.J.: Homological Properties of Finite Type Khovanov–Lauda–Rouquier Algebras. arXiv:1210.6900

  7. Cellini, P., Papi, P.: The structure of total reflection orders in affine root systems. J. Algebra 205, 207–226 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Grojnowski, I.: Affine \(\mathfrak{sl}_p\) Controls the Representation Theory of the Symmetric Group and Related Hecke Algebras. arXiv:math.RT/9907129

  9. Hill, D., Melvin, G., Mondragon, D.: Representations of quiver Hecke algebras via Lyndon bases. J. Pure Appl. Algebra 216, 1052–1079 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kac, V.G.: Infinite Dimensional Lie Algebras. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  11. Kang, S.-J., Kashiwara, M.: Categorification of highest weight modules via Khovanov–Lauda–Rouquier algebras. Invent. Math. 190, 699–742 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kashiwara, M.: Global crystal bases of quantum groups. Duke Math. J. 69, 455–485 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  13. Khovanov, M., Lauda, A.: A diagrammatic approach to categorification of quantum groups I. Represent. Theory 13, 309–347 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. Khovanov, M., Lauda, A.: A diagrammatic approach to categorification of quantum groups II. Trans. Am. Math. Soc. 363, 2685–2700 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kleshchev, A.: Linear and Projective Representations of Symmetric Groups. Cambridge University Press, Cambridge (2005)

    Book  MATH  Google Scholar 

  16. Kleshchev, A.: Imaginary Schur–Weyl duality (in preparation)

  17. Kleshchev, A., Ram, A.: Representations of KhovanovLauda Rouquier algebras and combinatorics of Lyndon words. Math. Ann. 349, 943–975 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kleshchev, A., Ram, A.: Homogeneous representations of KhovanovLauda algebras. J. Eur. Math. Soc. 12, 1293–1306 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  19. Lauda, A., Vazirani, M.: Crystals from categorified quantum groups. Adv. Math. 228, 803–861 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Leclerc, B.: Dual canonical bases, quantum shuffles and \(q\)-characters. Math. Z. 246, 691–732 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Lusztig, G.: Introduction to Quantum Groups. Birkhäuser, Basel (1993)

  22. Lusztig, G.: Braid group action and canonical bases. Adv. Math. 122, 237–261 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  23. McNamara, P.: Finite dimensional representations of Khovanov–Lauda–Rouquier algebras I: finite type. J. Reine Angew. Math. (to appear). arXiv:1207.5860

  24. Rouquier, R.: \(2\)-Kac-Moody algebras. arXiv:0812.5023

  25. Stembridge, J.R.: On the fully commutative elements of Coxeter groups. J. Algebraic Combin. 5, 353–385 (1996)

    MATH  MathSciNet  Google Scholar 

  26. Stembridge, J.R.: Minuscule elements of Weyl groups. J. Algebra 235, 722–743 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  27. Tingley, P., Webster, B.: Mirkovic–Vilonen polytopes and Khovanov–Lauda–Rouquier algebras. arXiv:1210.6921

  28. Varagnolo, M., Vasserot, E.: Canonical bases and KLR-algebras. J. Reine Angew. Math. 659, 67–100 (2011)

    MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

This paper has been influenced by the beautiful ideas of Peter McNamara [23], who also drew my attention to the paper [1] and suggested a slightly more general version of the main result appearing here after the first version of this paper was released. I am also grateful to Arun Ram and Jon Brundan for many useful conversations.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, OR, 97403, USA

    Alexander S. Kleshchev

Authors
  1. Alexander S. Kleshchev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander S. Kleshchev.

Additional information

Research supported in part by the NSF Grant No. DMS-1161094 and the Humboldt Foundation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kleshchev, A.S. Cuspidal systems for affine Khovanov–Lauda–Rouquier algebras. Math. Z. 276, 691–726 (2014). https://doi.org/10.1007/s00209-013-1219-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-013-1219-9

Mathematics Subject Classification (2000)

Search

Navigation