Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras

Shan, Peng

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 44 (2011) no. 1, pp. 147-182.

On définit les foncteurs de $i$-restriction et $i$-induction sur la catégorie $𝒪$ des algèbres de Hecke doublement affines rationnelles cyclotomiques. Ceci donne lieu à un cristal sur l’ensemble des classes d’isomorphismes de modules simples, qui est isomorphe au cristal d’un espace de Fock.

We define the $i$-restriction and $i$-induction functors on the category $𝒪$ of the cyclotomic rational double affine Hecke algebras. This yields a crystal on the set of isomorphism classes of simple modules, which is isomorphic to the crystal of a Fock space.

Classification : 20C08
Keywords: Hecke algebra, induction, restriction, crystal, Fock space, categorification
Mot clés : algèbre de Hecke, induction, restriction, cristal, espace de Fock, catégorification

[1] S. Ariki, On the decomposition numbers of the Hecke algebra of $G(m,1,n)$, J. Math. Kyoto Univ. 36 (1996), 789-808. | MR | Zbl

[2] S. Ariki, Representations of quantum algebras and combinatorics of Young tableaux, University Lecture Series 26, Amer. Math. Soc., 2002. | MR | Zbl

[3] A. Berenstein & D. Kazhdan, Geometric and unipotent crystals. II. From unipotent bicrystals to crystal bases, in Quantum groups, Contemp. Math. 433, Amer. Math. Soc., 2007, 13-88. | MR | Zbl

[4] R. Bezrukavnikov & P. Etingof, Parabolic induction and restriction functors for rational Cherednik algebras, Selecta Math. (N.S.) 14 (2009), 397-425. | Zbl

[5] M. Broué, G. Malle & J. Michel, Towards spetses. I, Transform. Groups 4 (1999), 157-218. | Zbl

[6] M. Broué, G. Malle & R. Rouquier, Complex reflection groups, braid groups, Hecke algebras, J. reine angew. Math. 500 (1998), 127-190. | Zbl

[7] J. Chuang & R. Rouquier, Derived equivalences for symmetric groups and ${\mathrm{𝔰𝔩}}_2$-categorification, Ann. of Math. 167 (2008), 245-298. | Zbl

[8] W. Crawley-Boevey & M. P. Holland, Noncommutative deformations of Kleinian singularities, Duke Math. J. 92 (1998), 605-635. | Zbl

[9] C. W. Curtis & I. Reiner, Methods of representation theory. Vol. II, with applications to finite groups and orders, Pure and Applied Mathematics, John Wiley & Sons Inc., 1987. | Zbl

[10] V. Ginzburg, N. Guay, E. Opdam & R. Rouquier, On the category $𝒪$ for rational Cherednik algebras, Invent. Math. 154 (2003), 617-651. | Zbl

[11] M. Jimbo, K. C. Misra, T. Miwa & M. Okado, Combinatorics of representations of $U_q\left(\widehat{\mathrm{𝔰𝔩}}\left(n\right)\right)$ at $q=0$, Comm. Math. Phys. 136 (1991), 543-566. | Zbl

[12] M. Kashiwara & T. Kawai, On holonomic systems of microdifferential equations. III. Systems with regular singularities, Publ. Res. Inst. Math. Sci. 17 (1981), 813-979. | Zbl

[13] S. Lyle & A. Mathas, Blocks of cyclotomic Hecke algebras, Adv. Math. 216 (2007), 854-878. | Zbl

[14] R. Rouquier, $q$-Schur algebras and complex reflection groups, Mosc. Math. J. 8 (2008), 119-158. | MR | Zbl

[15] R. Rouquier, 2-Kac-Moody algebras, preprint arXiv:0812.5023.

[16] D. Uglov, Canonical bases of higher-level $q$-deformed Fock spaces and Kazhdan-Lusztig polynomials, in Physical combinatorics (Kyoto, 1999), Progr. Math. 191, Birkhäuser, 2000, 249-299. | MR | Zbl

[17] X. Yvonne, A conjecture for $q$-decomposition matrices of cyclotomic $v$-Schur algebras, J. Algebra 304 (2006), 419-456. | MR | Zbl