Advertisement

Journal of Algebraic Combinatorics

, Volume 49, Issue 1, pp 99–124 | Cite as

Heisenberg algebra, wedges and crystals

  • Thomas GerberEmail author
Article
  • 62 Downloads

Abstract

We explain how the action of the Heisenberg algebra on the space of q-deformed wedges yields the Heisenberg crystal structure on charged multipartitions, by using the Boson–Fermion correspondence and looking at the action of the Schur functions at \(q=0\). In addition, we give the explicit formula for computing this crystal in full generality.

Keywords

Fock space Categorification Quantum groups Heisenberg algebra Crystals Symmetric functions Combinatorics 

Notes

Acknowledgements

Many thanks to Emily Norton for pointing out an inconsistency in the first version of this paper and for helpful conversations.

References

  1. 1.
    Ariki, S.: On the decomposition numbers of the Hecke algebra of \(G(m,1, n)\). J. Math. Kyoto Univ. 36(4), 789–808 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brundan, J., Kleshchev, A.: Graded decomposition numbers for cyclotomic Hecke algebras. Adv. Math. 222, 1883–1942 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dudas, O., Varagnolo, M., Vasserot, É.: Categorical actions on unipotent representations I. Finite unitary groups (2015). arXiv:1509.03269
  4. 4.
    Dudas, O., Varagnolo, M., Vasserot, É.: Categorical actions on unipotent representations of finite classical groups. In: Categorification and Higher Representation Theory. Contemporary Mathematics, vol. 683, pp. 41–104. American Mathematical Society, Providence, RI (2017)Google Scholar
  5. 5.
    Etingof, P.: Supports of irreducible spherical representations of rational Cherednik algebras of finite Coxeter groups. Adv. Math. 229, 2042–2054 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Foda, O., Leclerc, B., Okado, M., Thibon, J.-Y., Welsh, T.: Branching functions of \(A_{n-1}^{(1)}\) and Jantzen–Seitz problem for Ariki–Koike algebras. Adv. Math. 141, 322–365 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Geck, M., Jacon, N.: Representations of Hecke Algebras at Roots of Unity. Springer, Berlin (2011)CrossRefzbMATHGoogle Scholar
  8. 8.
    Gerber, T.: Crystal isomorphisms in Fock spaces and Schensted correspondence in affine type A. Algebra Rep. Theory 18, 1009–1046 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gerber, T.: Triple crystal action in Fock spaces. Adv. Math. (2016, to appear).  https://doi.org/10.1016/j.aim.2018.02.030
  10. 10.
    Gerber, T., Hiss, G., Jacon, N.: Harish–Chandra series in finite unitary groups and crystal graphs. Int. Math. Res. Not. 22, 12206–12250 (2015)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Iijima, K.: On a higher level extension of Leclerc–Thibon product theorem in \(q\)-deformed Fock spaces. J. Algebra 371, 105–131 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jacon, N., Lecouvey, C.: A combinatorial decomposition of higher level Fock spaces. Osaka J. Math. 50(4), 897–920 (2013)MathSciNetzbMATHGoogle Scholar
  13. 13.
    James, G., Kerber, A.: The Representation Theory of the Symmetric Group. Cambridge University Press, Cambridge (1984)CrossRefGoogle Scholar
  14. 14.
    Jimbo, M., Misra, K.C., Miwa, T., Okado, M.: Combinatorics of representations of \(U_q(\widehat{sl(n)})\) at \(q=0\). Commun. Math. Phys. 136(3), 543–566 (1991)CrossRefzbMATHGoogle Scholar
  15. 15.
    Kashiwara, M.: Global crystal bases of quantum groups. Duke Math. J. 69, 455–485 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kashiwara, M., Miwa, T., Stern, E.: Decomposition of \(q\)-deformed Fock spaces. Sel. Math. 1, 787–805 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lascoux, A., Leclerc, B., Thibon, J.-Y.: Ribbon Tableaux, Hall–Littlewood Functions, quantum affine algebras and unipotent varieties. J. Math. Phys. 38, 1041–1068 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Leclerc, B., Thibon, J.-Y.: Littlewood–Richardson coefficients and Kazhdan–Lusztig polynomials. In: Combinatorial Methods in Representation Theory, volume 28 of Advanced Studies in Pure Mathematics. American Mathematical Society (2001)Google Scholar
  19. 19.
    Losev, I.: Supports of simple modules in cyclotomic Cherednik categories O (2015). arXiv:1509.00526
  20. 20.
    Lusztig, G.: Modular representations and quantum groups. Contemp. Math. 82, 58–77 (1989)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford Mathematical Monographs, Oxford (1998)zbMATHGoogle Scholar
  22. 22.
    Miwa, T., Jimbo, M., Date, E.: Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  23. 23.
    Shan, P.: Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras. Ann. Sci. Éc. Norm. Supér. 44, 147–182 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Shan, P., Vasserot, É.: Heisenberg algebras and rational double affine Hecke algebras. J. Am. Math. Soc. 25, 959–1031 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge University Press, Cambridge (2001)zbMATHGoogle Scholar
  26. 26.
    Tingley, P.: Three combinatorial models for \(\widehat{{\mathfrak{s}}_{n}}\) crystals, with applications to cylindric plane partitions. Int. Math. Res. Not. 143, 1–40 (2008)Google Scholar
  27. 27.
    Uglov, D.: Canonical bases of higher-level \(q\)-deformed Fock spaces and Kazhdan–Lusztig polynomials. Prog. Math. 191, 249–299 (1999)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Yvonne, X.: Bases canoniques d’espaces de Fock de niveau supérieur. Ph.D. thesis, Université de Caen (2005)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Lehrstuhl D für MathematikRWTH Aachen UniversityAachenGermany

Personalised recommendations