On the uniqueness of promotion operators on tensor products of type A crystals
Jason Bandlow
, Anne Schilling
and Nicolas M. Thiéry
DOI: 10.1007/s10801-009-0182-3
Abstract
The affine Dynkin diagram of type A n (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 A n 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 A n 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.
Pages: 217–251
Keywords: keywords affine crystal bases; promotion operator; Schur polynomial factorization
Full Text: PDF
References
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3. Fulton, W.: Young tableaux. London Mathematical Society Student Texts, vol.
35. Cambridge University Press, Cambridge (1997). With applications to representation theory and geometry
4. Haiman, M.D.: Dual equivalence with applications, including a conjecture of Proctor. Discrete Math. 99(1-3), 79-113 (1992)
5. Hernandez, D.: Quantum toroidal algebras and their representations. Preprint (2008).
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).
7. Kashiwara, M.: Crystal bases of modified quantized enveloping algebra. Duke Math. J. 73(2), 383- 413 (1994)
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)
9. Kashiwara, M.: On level-zero representations of quantized affine algebras. Duke Math. J. 112(1), 117-175 (2002)
10. Kashiwara, M.: Level zero fundamental representations over quantized affine algebras and Demazure modules. Publ. Res. Inst. Math. Sci. 41(1), 223-250 (2005)
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)
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)
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)
14. Kleber, M.: Plücker relations on Schur functions. J. Algebraic Combin. 13(2), 199-211 (2001)
15. Kashiwara, M., Nakashima, T.: Crystal graphs for representations of the q-analogue of classical Lie algebras. J. Algebra 165(2), 295-345 (1994)
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)
17. Lascoux, A., Leclerc, B., Thibon, J.-Y.: Crystal graphs and q-analogues of weight multiplicities for the root system An. Lett. Math. Phys. 35(4), 359-374 (1995)
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)
19. Okado, M., Schilling, A.: Existence of Kirillov-Reshetikhin crystals for nonexceptional types. Represent. Theory 12, 186-207 (2008). [math.QA]
20. Petersen, T.K., Pylyavskyy, P., Rhoades, B.: Promotion and cyclic sieving via webs. Preprint (2008).
21. Purbhoo, K., van Willigenburg, S.: On tensor products of polynomial representations. Canad. Math. Bull. 51(4), 584-592 (2008)
22. Rajan, C.S.: Unique decomposition of tensor products of irreducible representations of simple algebraic groups. Ann. Math. (2) 160(2), 683-704 (2004)
23. Rhoades, B.: Cyclic sieving and promotion. Preprint (2008)
24. Stein, W.A., et al.: Sage Mathematics Software (Version 3.3). The Sage Development Team, 2009.