×

zbMATH — the first resource for mathematics

Graded decomposition numbers for cyclotomic Hecke algebras. (English) Zbl 1241.20003
From the introduction: In recent joint work with W. Wang [J. Reine Angew. Math. 655, 61-87 (2011; Zbl 1244.20003)], we have constructed graded Specht modules for cyclotomic Hecke algebras. In this article, we prove a graded version of the Lascoux-Leclerc-Thibon conjecture, describing the decomposition numbers of graded Specht modules over a field of characteristic zero.
We end the introduction with a brief guide to the rest of the article, indicating some of the other things to be found here. Section 2 is primarily devoted to recalling the definition of the algebras \(R_d\) in type \(\widehat{\mathfrak{sl}}_e\), and then reviewing some of the foundational results proved about them by M. Khovanov and A. D. Lauda [in Represent. Theory 13, 309-347 (2009; Zbl 1188.81117)].
In Section 3 we review the construction of the irreducible highest weight module \(V(\Lambda)\) over \(U_q(\widehat{\mathfrak{sl}}_e)\) as a summand of Fock space. At the same time, we construct various bases for these modules, paralleling the setup of J. Brundan and A. Kleshchev [Math. Z. 266, No. 4, 877-919 (2010; Zbl 1287.17022), §2] closely. This part of the story is surprisingly lengthy as there are some subtle combinatorial issues surrounding the triangularity of the standard monomials in \(V(\Lambda)\); see Section 3.9. Unlike almost all of the literature in the subject, our approach emphasizes the dual-canonical basis rather than the canonical basis.
In Section 4 we consider the cyclotomic quotients \(R^\Lambda_d\) of \(R_d\) introduced originally by M. Khovanov and A. D. Lauda [in loc. cit., §3.4]. We use the isomorphism between \(R^\Lambda_d\) and \(H^\Lambda_d\) from J. Brundan and A. Kleshchev [Invent. Math. 178, No. 3, 451-484 (2009; Zbl 1201.20004)] to quickly deduce the classification of irreducible graded \(R^\Lambda_d\)-modules from I. Grojnowski’s classification of irreducible \(H^\Lambda_d\) in terms of crystal graphs from [Affine \(\mathfrak{sl}_p\) controls the representation theory of the symmetric group and related Hecke algebras, arXiv:math.RT/9907129.G2]; see Section 4.8. At the same time we lift various branching rules to the graded setting. Then we prove the first key categorification theorem, which identifies \(V(\Lambda)\) with the direct sum \(\bigoplus_{d\geq 0}[\text{Proj}(R^\Lambda_d)]_{\mathbb Q(q)}\) as above; see Section 4.10. As an application, we compute the graded dimension of \(R^\Lambda_d\); see Section 4.11. We stress that this part of the development makes sense over any ground field, and does not depend on any results from geometric representation theory.
In Section 5 we lift Ariki’s results to the graded setting to prove simultaneously the graded version of the Lascoux-Leclerc-Thibon conjecture and the Khovanov-Lauda conjecture; see Section 5.5. In the course of this we encounter some non-trivial issues related to the parametrization of irreducible modules: there are two relevant parametrizations, one arising from the crystal graph and the other arising from Specht module theory; see Section 5.4 for the latter. The identification of the two parametrizations is addressed in Ariki’s work, but we give a self-contained treatment here in order to keep track of gradings. We also discuss the situation over fields of positive characteristic, introducing graded analogues of James’ adjustment matrices; see Section 5.6.

MSC:
20C08 Hecke algebras and their representations
17B37 Quantum groups (quantized enveloping algebras) and related deformations
20C30 Representations of finite symmetric groups
20C20 Modular representations and characters
20F55 Reflection and Coxeter groups (group-theoretic aspects)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Ariki, S., On the decomposition numbers of the Hecke algebra of \(G(m, 1, n)\), J. math. Kyoto univ., 36, 789-808, (1996) · Zbl 0888.20011
[2] Ariki, S., On the classification of simple modules for the cyclotomic Hecke algebra of type \(G(m, 1, n)\) and kleshchev multipartitions, Osaka J. math., 38, 827-837, (2001) · Zbl 1005.20007
[3] Ariki, S., Representations of quantum algebras and combinatorics of Young tableaux, Univ. lecture ser., vol. 26, (2002), American Mathematical Society Providence, RI · Zbl 1003.17008
[4] Ariki, S., Proof of the modular branching rule for cyclotomic Hecke algebras, J. algebra, 306, 290-300, (2006) · Zbl 1130.20005
[5] Ariki, S.; Koike, K., A Hecke algebra of \((\mathbb{Z} / r \mathbb{Z}) \wr S_n\) and construction of its irreducible representations, Adv. math., 106, 216-243, (1994) · Zbl 0840.20007
[6] Ariki, S.; Mathas, A.; Rui, H., Cyclotomic Nazarov-wenzl algebras, Nagoya math. J., 182, 47-134, (2006) · Zbl 1159.20008
[7] Beilinson, A.; Ginzburg, V.; Soergel, W., Koszul duality patterns in representation theory, J. amer. math. soc., 9, 473-527, (1996) · Zbl 0864.17006
[8] Bernstein, J.; Zelevinsky, A., Induced representations of reductive p-adic groups I, Ann. sci. école norm. sup., 10, 441-472, (1977) · Zbl 0412.22015
[9] Brundan, J., Modular branching rules and the Mullineux map for Hecke algebras of type A, Proc. London math. soc., 77, 551-581, (1998)
[10] Brundan, J., Dual canonical bases and Kazhdan-Lusztig polynomials, J. algebra, 306, 17-46, (2006) · Zbl 1169.17008
[11] Brundan, J., Centers of degenerate cyclotomic Hecke algebras and parabolic category \(\mathcal{O}\), Represent. theory, 12, 236-259, (2008) · Zbl 1202.20008
[12] Brundan, J.; Kleshchev, A., Representation theory of symmetric groups and their double covers, (), 31-53 · Zbl 1043.20005
[13] Brundan, J.; Kleshchev, A., Representations of shifted Yangians and finite W-algebras, Mem. amer. math. soc., 196, 918, 107, (2008) · Zbl 1169.17009
[14] Brundan, J.; Kleshchev, A., Schur-Weyl duality for higher levels, Selecta math. (N.S.), 14, 1-57, (2008) · Zbl 1211.17012
[15] Brundan, J.; Kleshchev, A., Blocks of cyclotomic Hecke algebras and Khovanov-lauda algebras, Invent. Math., in press · Zbl 1201.20004
[16] Brundan, J.; Kleshchev, A., The degenerate analogue of Ariki’s categorification theorem · Zbl 1287.17022
[17] Brundan, J.; Kleshchev, A.; Wang, W., Graded Specht modules · Zbl 1244.20003
[18] Brundan, J.; Stroppel, C., Highest weight categories arising from Khovanov’s diagram algebra III: category \(\mathcal{O}\) · Zbl 1261.17006
[19] Chriss, N.; Ginzburg, V., Representation theory and complex geometry, (1997), Birkhäuser · Zbl 0879.22001
[20] Chuang, J.; Rouquier, R., Derived equivalences for symmetric groups and \(\mathfrak{sl}_2\)-categorification, Ann. of math., 167, 245-298, (2008) · Zbl 1144.20001
[21] Dipper, R.; James, G.D.; Mathas, A., Cyclotomic q-Schur algebras, Math. Z., 229, 385-416, (1998) · Zbl 0934.20014
[22] Donkin, S., The q-Schur algebra, (1998), Cambridge University Press Cambridge · Zbl 0927.20003
[23] Foda, O.; Leclerc, B.; Okado, M.; Thibon, J.-Y.; Welsh, T., Branching functions of \(A_n^{(1)}\) and jantzen-seitz problem for ariki-Koike algebras, Adv. math., 141, 322-365, (1999) · Zbl 0930.17023
[24] Ford, B.; Kleshchev, A., A proof of the Mullineux conjecture, Math. Z., 226, 267-308, (1997) · Zbl 0958.20018
[25] Graham, J.J.; Lehrer, G.I., Cellular algebras, Invent. math., 123, 1-34, (1996) · Zbl 0853.20029
[26] Grojnowski, I., Representations of affine Hecke algebras (and affine quantum \(\mathit{GL}_n\)) at roots of unity, Int. math. res. not., 5, 215-217, (1994) · Zbl 0819.17009
[27] Grojnowski, I., Affine \(\mathfrak{sl}_p\) controls the representation theory of the symmetric group and related Hecke algebras · Zbl 0819.17009
[28] Hayashi, T., q-analogues of Clifford and Weyl algebras: spinor and oscillator representations of quantum enveloping algebras, Comm. math. phys., 127, 129-144, (1990) · Zbl 0701.17008
[29] Jimbo, M.; Misra, K.; Miwa, T.; Okado, M., Combinatorics of representations of \(U_q(\hat{\mathfrak{sl}}(n))\) at \(q = 0\), Comm. math. phys., 136, 543-566, (1991) · Zbl 0749.17015
[30] Kac, V.G., Infinite dimensional Lie algebras, (1990), Cambridge University Press · Zbl 0425.17009
[31] Kashiwara, M., Crystallizing the q-analogue of universal enveloping algebras, Comm. math. phys., 133, 249-260, (1990) · Zbl 0724.17009
[32] Kashiwara, M., On crystal bases of the q-analogue of universal enveloping algebras, Duke math. J., 63, 465-516, (1991) · Zbl 0739.17005
[33] Kashiwara, M., Global crystal bases of quantum groups, Duke math. J., 69, 455-485, (1993) · Zbl 0774.17018
[34] Kashiwara, M., On crystal bases, (), 155-197 · Zbl 0851.17014
[35] Kazhdan, D.; Lusztig, G., Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. math., 87, 153-215, (1987) · Zbl 0613.22004
[36] Khovanov, M.; Lauda, A., A diagrammatic approach to categorification of quantum groups I · Zbl 1188.81117
[37] Khovanov, M.; Lauda, A., A diagrammatic approach to categorification of quantum groups II · Zbl 1214.81113
[38] Khovanov, M.; Lauda, A., A diagrammatic approach to categorification of quantum groups III · Zbl 1214.81113
[39] Kleshchev, A., Branching rules for modular representations of symmetric groups II, J. reine angew. math., 459, 163-212, (1995) · Zbl 0817.20009
[40] Kleshchev, A., Branching rules for modular representations of symmetric groups III: some corollaries and a problem of Mullineux, J. London math. soc., 54, 25-38, (1996) · Zbl 0854.20014
[41] Kleshchev, A., Linear and projective representations of symmetric groups, (2005), Cambridge University Press Cambridge · Zbl 1080.20011
[42] Lascoux, A.; Leclerc, B.; Thibon, J.-Y., Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. math. phys., 181, 205-263, (1996) · Zbl 0874.17009
[43] Leclerc, B., Dual canonical bases, quantum shuffles and q-characters, Math. Z., 246, 691-732, (2004) · Zbl 1052.17008
[44] Leclerc, B.; Thibon, J.-Y., Canonical bases of q-deformed Fock spaces, Int. math. res. not., 9, 447-456, (1996) · Zbl 0863.17013
[45] Leclerc, B.; Thibon, J.-Y., Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials, (), 155-220 · Zbl 1058.20006
[46] Lusztig, G., Introduction to quantum groups, (1993), Birkhäuser · Zbl 0788.17010
[47] Lyle, S.; Mathas, A., Blocks of cyclotomic Hecke algebras, Adv. math., 216, 854-878, (2007) · Zbl 1156.20006
[48] Malle, G.; Mathas, A., Symmetric cyclotomic Hecke algebras, J. algebra, 205, 275-293, (1998) · Zbl 0913.20006
[49] Mathas, A., Tilting modules for cyclotomic Schur algebras, J. reine angew. math., 562, 137-169, (2003) · Zbl 1065.20009
[50] Misra, K.C.; Miwa, T., Crystal base of the basic representation of \(U_q(\hat{\mathfrak{sl}}_n)\), Comm. math. phys., 134, 79-88, (1990) · Zbl 0724.17010
[51] Nǎstǎsescu, C.; Van Oystaeyen, F., Methods of graded rings, Lecture notes in math., vol. 1836, (2004), Springer · Zbl 1043.16017
[52] Rickard, J., Equivalences of derived categories for symmetric algebras, J. algebra, 257, 460-481, (2002) · Zbl 1033.20005
[53] Rouquier, R., q-Schur algebras and complex reflection groups, Mosc. math. J., 8, 119-158, (2008) · Zbl 1213.20007
[54] Rouquier, R., 2-Kac-Moody algebras
[55] Stern, E., Semi-infinite wedges and vertex operators, Int. math. res. not., 201-220, (1995) · Zbl 0823.17042
[56] Takemura, K.; Uglov, D., Representations of the quantum toroidal algebra on highest weight modules of the quantum affine algebra of type \(\mathfrak{gl}_N\), Publ. res. inst. math. sci., 35, 407-450, (1999) · Zbl 0974.17019
[57] Turner, W., Rock blocks, Mem. Amer. Math. Soc., in press · Zbl 1269.20010
[58] Uglov, D., Canonical bases of higher level q-deformed Fock spaces and Kazhdan-Lusztig polynomials, (), 249-299 · Zbl 0963.17012
[59] Varagnolo, M.; Vasserot, E., On the decomposition matrices of the quantized Schur algebra, Duke math. J., 100, 267-297, (1999) · Zbl 0962.17006
[60] Varagnolo, M.; Vasserot, E., Cyclotomic double affine Hecke algebras and affine parabolic category \(\mathcal{O}\) · Zbl 1202.20009
[61] Varagnolo, M.; Vasserot, E., Canonical bases and Khovanov-lauda algebras · Zbl 1229.17019
[62] Vazirani, M., Parameterizing Hecke algebra modules: Bernstein-Zelevinsky multisegments, kleshchev multipartitions, and crystal graphs, Transform. groups, 7, 267-303, (2002) · Zbl 1061.20007
[63] Yvonne, X., A conjecture for q-decomposition numbers of cyclotomic v-Schur algebras, J. algebra, 304, 419-456, (2006) · Zbl 1130.20009
[64] Yvonne, X., Canonical bases of higher-level q-deformed Fock spaces, J. algebraic combin., 26, 383-414, (2007) · Zbl 1223.17018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.