Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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CYCLOTOMIC QUIVER HECKE ALGEBRAS CORRESPONDING TO MINUSCULE REPRESENTATIONS

  • Received : 2019.09.23
  • Accepted : 2020.06.10
  • Published : 2020.11.01
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Abstract

In the paper, we give an explicit basis of the cyclotomic quiver Hecke algebra corresponding to a minuscule representation of finite type.

Keywords

Acknowledgement

The research was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (NRF-2017R1A1A1A05001058).

References

  1. S. Ariki, K. Iijima, and E. Park, Representation type of finite quiver Hecke algebras of type $A_{\ell}^{(1)}$ for arbitrary parameters, Int. Math. Res. Not. IMRN 2015, no. 15, 6070-6135. https://doi.org/10.1093/imrn/rnu115
  2. S. Ariki and E. Park, Representation type of finite quiver Hecke algebras of type $A_{2\ell}^{(2)}$, J. Algebra 397 (2014), 457-488. https://doi.org/10.1016/j.jalgebra.2013.09.005
  3. N. Bourbaki, Lie groups and Lie algebras. Chapters 7-9, translated from the 1975 and 1982 French originals by Andrew Pressley, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2005.
  4. J. Brundan and A. Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, Invent. Math. 178 (2009), no. 3, 451-484. https://doi.org/10.1007/s00222-009-0204-8
  5. G. Feinberg and K.-H. Lee, Fully commutative elements of type D and homogeneous representations of KLR-algebras, J. Comb. 6 (2015), no. 4, 535-557. https://doi.org/10.4310/JOC.2015.v6.n4.a7
  6. G. Feinberg and K.-H. Lee, Homogeneous representations of type A KLR-algebras and Dyck paths, Sem. Lothar. Combin. 75 ([2015-2019]), Art. B75b, 19 pp.
  7. M. Geck and G. Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs. New Series, 21, The Clarendon Press, Oxford University Press, New York, 2000.
  8. J. Hong and S.-J. Kang, Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, 42, American Mathematical Society, Providence, RI, 2002. https://doi.org/10.1090/gsm/042
  9. M. Jimbo and T. Miwa, Solitons and infinite-dimensional Lie algebras, Publ. Res. Inst. Math. Sci. 19 (1983), no. 3, 943-1001. https://doi.org/10.2977/prims/1195182017
  10. S.-J. Kang and M. Kashiwara, Categorification of highest weight modules via Khovanov- Lauda-Rouquier algebras, Invent. Math. 190 (2012), no. 3, 699-742. https://doi.org/10.1007/s00222-012-0388-1
  11. M. Kashiwara, Crystalizing the q-analogue of universal enveloping algebras, Comm. Math. Phys. 133 (1990), no. 2, 249-260. http://projecteuclid.org/euclid.cmp/1104201397 https://doi.org/10.1007/BF02097367
  12. M. Kashiwara, On crystal bases of the Q-analogue of universal enveloping algebras, Duke. Math. J. 63 (1991), no. 2, 46516. https://doi.org/10.1215/S0012-7094-91-06321-0
  13. M. Kashiwara, Biadjointness in cyclotomic Khovanov-Lauda-Rouquier algebras, Publ. Res. Inst. Math. Sci. 48 (2012), no. 3, 501-524. https://doi.org/10.2977/PRIMS/78
  14. M. Kashiwara, M. Kim, S. Oh, and E. Park, Monoidal categories associated with strata of flag manifolds, Adv. Math. 328 (2018), 959-1009. https://doi.org/10.1016/j.aim.2018.02.013
  15. M. Khovanov and A. D. Lauda, A diagrammatic approach to categorification of quantum groups. I, Represent. Theory 13 (2009), 309-347. https://doi.org/10.1090/S1088-4165-09-00346-X
  16. M. Khovanov and A. D. Lauda, A diagrammatic approach to categorification of quantum groups II, Trans. Amer. Math. Soc. 363 (2011), no. 5, 2685-2700. https://doi.org/10.1090/S0002-9947-2010-05210-9
  17. A. Kleshchev and A. Ram, Homogeneous representations of Khovanov-Lauda algebras, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 5, 1293-1306. https://doi.org/10.4171/JEMS/230
  18. R. Rouquier, 2 Kac-Moody algebras, arXiv:0812.5023 (2008).
  19. J. R. Stembridge, Minuscule elements of Weyl groups, J. Algebra 235 (2001), no. 2, 722-743. https://doi.org/10.1006/jabr.2000.8488