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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.

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)
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