Journal of Algebraic Combinatorics

, Volume 31, Issue 2, pp 217–251 | Cite as

On the uniqueness of promotion operators on tensor products of type A crystals

  • Jason Bandlow
  • Anne Schilling
  • Nicolas M. Thiéry
Open Access
Article

Abstract

The affine Dynkin diagram of type An(1) has a cyclic symmetry. The analogue of this Dynkin diagram automorphism on the level of crystals is called a promotion operator. In this paper we show that the only irreducible type An crystals which admit a promotion operator are the highest weight crystals indexed by rectangles. In addition we prove that on the tensor product of two type An crystals labeled by rectangles, there is a single connected promotion operator. We conjecture this to be true for an arbitrary number of tensor factors. Our results are in agreement with Kashiwara’s conjecture that all ‘good’ affine crystals are tensor products of Kirillov-Reshetikhin crystals.

Keywords

Affine crystal bases Promotion operator Schur polynomial factorization 

References

  1. 1.
    Akasaka, T., Kashiwara, M.: Finite-dimensional representations of quantum affine algebras. Publ. Res. Inst. Math. Sci. 33(5), 839–867 (1997) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Deka, L., Schilling, A.: New fermionic formula for unrestricted Kostka polynomials. J. Combin. Theory Ser. A 113(7), 1435–1461 (2006) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Fulton, W.: Young tableaux. London Mathematical Society Student Texts, vol. 35. Cambridge University Press, Cambridge (1997). With applications to representation theory and geometry MATHGoogle Scholar
  4. 4.
    Haiman, M.D.: Dual equivalence with applications, including a conjecture of Proctor. Discrete Math. 99(1–3), 79–113 (1992) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Hernandez, D.: Quantum toroidal algebras and their representations. Preprint (2008). arXiv:0801.2397
  6. 6.
    Hivert, F., Thiéry, N.M.: MuPAD-Combinat, an open-source package for research in algebraic combinatorics. Sém. Lothar. Combin. 51:Art. B51z, 70 pp. (electronic) (2004). http://mupad-combinat.sf.net/
  7. 7.
    Kashiwara, M.: Crystal bases of modified quantized enveloping algebra. Duke Math. J. 73(2), 383–413 (1994) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kashiwara, M.: On crystal bases. In: Representations of groups, Banff, AB, 1994. CMS Conf. Proc., vol. 16, pp. 155–197. Am. Math. Soc., Providence (1995) Google Scholar
  9. 9.
    Kashiwara, M.: On level-zero representations of quantized affine algebras. Duke Math. J. 112(1), 117–175 (2002) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kashiwara, M.: Level zero fundamental representations over quantized affine algebras and Demazure modules. Publ. Res. Inst. Math. Sci. 41(1), 223–250 (2005) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kirillov, A.N.: Combinatorial identities and completeness of states of the Heisenberg magnet. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 131, 88–105 (1983). Questions in quantum field theory and statistical physics, 4, transl. in J. Soviet Math. 36, 115–128 (1987) MathSciNetGoogle Scholar
  12. 12.
    Kang, S.-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T., Nakayashiki, A.: Perfect crystals of quantum affine Lie algebras. Duke Math. J. 68(3), 499–607 (1992) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kerov, S.V., Kirillov, A.N., Reshetikhin, N.Yu.: Combinatorics, the Bethe ansatz and representations of the symmetric group. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 155, 50–64, 193 (1986) (Differentsialnaya Geometriya, Gruppy Li i Mekh. VIII) Google Scholar
  14. 14.
    Kleber, M.: Plücker relations on Schur functions. J. Algebraic Combin. 13(2), 199–211 (2001) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kashiwara, M., Nakashima, T.: Crystal graphs for representations of the q-analogue of classical Lie algebras. J. Algebra 165(2), 295–345 (1994) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kirillov, A.N., Schilling, A., Shimozono, M.: A bijection between Littlewood-Richardson tableaux and rigged configurations. Selecta Math. (N.S.) 8(1), 67–135 (2002) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Lascoux, A., Leclerc, B., Thibon, J.-Y.: Crystal graphs and q-analogues of weight multiplicities for the root system A n. Lett. Math. Phys. 35(4), 359–374 (1995) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lascoux, A., Leclerc, B., Thibon, J.-Y.: The plactic monoid. Preliminary draft of a chapter for the new Lothaire book “Algebraic Combinatorics on Word”, 29 p. (1997) Google Scholar
  19. 19.
    Okado, M., Schilling, A.: Existence of Kirillov-Reshetikhin crystals for nonexceptional types. Represent. Theory 12, 186–207 (2008). arXiv:0706.2224v2 [math.QA] MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Petersen, T.K., Pylyavskyy, P., Rhoades, B.: Promotion and cyclic sieving via webs. Preprint (2008). arXiv:0804.3375
  21. 21.
    Purbhoo, K., van Willigenburg, S.: On tensor products of polynomial representations. Canad. Math. Bull. 51(4), 584–592 (2008) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Rajan, C.S.: Unique decomposition of tensor products of irreducible representations of simple algebraic groups. Ann. Math. (2) 160(2), 683–704 (2004) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Rhoades, B.: Cyclic sieving and promotion. Preprint (2008) Google Scholar
  24. 24.
    Stein, W.A., et al.: Sage Mathematics Software (Version 3.3). The Sage Development Team, 2009. http://www.sagemath.org
  25. 25.
    Schützenberger, M.P.: Promotion des morphismes d’ensembles ordonnés. Discrete Math. 2, 73–94 (1972) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Schützenberger, M.-P.: La correspondance de Robinson. In: Combinatoire et représentation du groupe symétrique, Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976. Lecture Notes in Math., vol. 579, pp. 59–113. Springer, Berlin (1977) CrossRefGoogle Scholar
  27. 27.
    Schilling, A.: Crystal structure on rigged configurations. Int. Math. Res. Not., Art. ID 97376, 27 (2006) Google Scholar
  28. 28.
    Shimozono, M.: Affine type A crystal structure on tensor products of rectangles, Demazure characters, and nilpotent varieties. J. Algebraic Combin. 15(2), 151–187 (2002) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Stembridge, J.R.: Multiplicity-free products of Schur functions. Ann. Comb. 5(2), 113–121 (2001) MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Stembridge, J.R.: A local characterization of simply-laced crystals. Trans. Amer. Math. Soc. 355(12), 4807–4823 (2003) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  • Jason Bandlow
    • 1
  • Anne Schilling
    • 2
  • Nicolas M. Thiéry
    • 3
    • 4
  1. 1.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA
  2. 2.Department of MathematicsUniversity of CaliforniaDavisUSA
  3. 3.Univ Paris-SudLaboratoire de Mathématiques d’OrsayOrsayFrance
  4. 4.CNRSOrsayFrance

Personalised recommendations