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We study the blocks for Ariki-Koike algebras using a general notion of core for $l$-partitions. We interpret the action of the affine symmetric group on the blocks in the context of level rank duality and study the orbits under this action.

We compare two generalisations of the notion of hook lengths for partitions. We apply this in the context of the modular representation theory of Ariki-Koike algebras. We show that the Schur element of a simple module is divisible by the Schur element of the associated (generalised) core. In the case of Hecke algebras of type $A$, we obtain an even stronger result: the Schur element of a simple module is equal to the product of the Schur element of its core and the Schur element of its quotient., Comment: 13 pages

We present a new combinatorial and conjectural algorithm for computing the Mullineux involution for the symmetric group and its Hecke algebra. This algorithm is built on a conjectural property of crystal isomorphisms which can be rephrased in a purely combinatorial way.

We study the relationship between Calogero-Moser cellular characters and characters defined from vectors of a Fock space of type $A_{\infty}$. Using this interpretation, we show that Lusztig's constructible characters of the Weyl group of type $B$ are sums of Calogero-Moser cellular characters. We also give an explicit construction of the character of minimal $b$-invariant of a given Calogero-Moser family of the complex reflection group $G(l,1,n)$., Comment: 19 pages, comments welcome

We develop a new approach for the computation of the Mullineux involution for the symmetric group and its Hecke algebra using the notion of crystal isomorphism and the Iwahori-Matsumoto involution for the affine Hecke algebra of type A. As a consequence, we obtain several new elementary combinatorial algorithms for its computation, one of which is equivalent to Xu's algorithm (and thus Mullineux' original algorithm). We thus obtain a simple interpretation of these algorithms and a new elementary proof that they indeed compute the Mullineux involution.

The complexity of a block of a symmetric algebra can be measured by the notion of defect, a numerical datum associated to each of the simple modules contained in the block. Geck showed that the defect is a block invariant for Iwahori-Hecke algebras of finite Coxeter groups in the equal parameter case, and speculated that a similar result should hold in the unequal parameter case. We conjecture that the defect is a block invariant for all cyclotomic Hecke algebras associated with complex reflection groups, and we prove it for the groups of type $G(l,p,n)$ and for the exceptional types for which the blocks are known. In particular, for the groups $G(l,1,n)$, we show that the defect corresponds to the notion of weight in the sense of Fayers, for which we thus obtain a new way of computation. We also prove that the defect is a block invariant for cyclotomic Yokonuma-Hecke algebras., Comment: 25 pages

We study the modular representation theory of the symmetric and alternating groups. One of the most natural ways to label the irreducible representations of a given group or algebra in the modular case is to show the unitriangularity of the decomposition matrices, that is, the existence of a unitriangular basic set. We study several ways to obtain such sets in the general case of a symmetric algebra. We apply our results to the symmetric groups and to their Hecke algebras and thus obtain new ways to label the simple modules for these objects. Finally, we show that these sets do not always exist in the case of the alternating groups by studying two explicit cases in characteristic 3.

We study a natural generalization of the notion of cores for l-partitions attached with a multi-charge s $\in$ Z^l : the (e, s)-cores. We rely them both to the combinatorics and the notion of weight defined by Fayers. Next we study applications in the context of the block theory for Ariki-Koike algebras.

The Key map is an important tool in the determination of the Demazure crystals associated to Kac-Moody algebras. In finite type A, it can be computed in the tableau realization of crystals by a simple combinatorial procedure due to Lascoux and Sch{\"u}tzenberger. We show that this procedure is a part of a more general construction holding in the Kac-Moody case that we illustrate in finite types and affine type A. In affine type A, we introduce higher level generalizations of core partitions which notably give interesting analogues of the Young lattice and are expected to parametrize distinguished elements of certain remarkable blocks for Ariki-Koike algebras.

We study the class of Uglov bipartitions and prove a generalization of a conjecture by Dipper, James and Murphy. We give two consequences concerning the computation of canonical bases in affine type A and the description of decomposition matrices for Hecke algebras of type Bn in arbitrary characteristic.

We define a generalization of the Mullineux involution on multipartitions using the theory of crystals for higher level Fock spaces. Our generalized Mullineux involution turns up in representation theory via two important derived functors on cyclotomic Cherednik category $\mathcal{O}$: Losev's "$\kappa=0$" wall-crossing, and the Ringel duality., Comment: Final version. Proof of Proposition 3.3, Lemma 3.10 and Lemma 3.13 rewritten in more detail

We give a new simple characterization of the set of Kleshchev multipartitions, and more generally of the set of Uglov multipartitions. These combinatorial objects play an important role in various areas of representation theory of quantum groups, Hecke algebras or finite reductive groups. As a consequence, we obtain a proof of a generalization of a conjecture by Dipper, James and Murphy and a generalization of the LLT algorithm for arbitrary level., Comment: Correction of a gap in the proof of the main Theorem. The Theorem is generalized to all types of Uglov multipartitions. Final version

We study the effect of Alvis-Curtis duality on the unipotent representations of $\mathrm{GL}_n(q)$ in non-defining characteristic $\ell$. We show that the permutation induced on the simple modules can be expressed in terms of a generalization of the Mullineux involution on the set of all partitions, which involves both $\ell$ and the order of $q$ modulo $\ell$.

We show that the wall crossing bijections between simples of the category O of the rational Cherednik algebras reduce to particular crystal isomorphisms which can be computed by a simple combinatorial procedure on multipartitions of fixed rank.

We study the natural labeling of the one dimensional representations for Ariki-Koike algebras at roots of unity. For Hecke algebras of types A and B, some of these representations can be identified with the socle of the Steinberg representation of a finite reductive group. We here give closed formulas for them. This uses, in particular, several results concerning crystal isomorphisms and the Mullineux involution.

The distribution of the unipotent modules (in non-defining prime characteristic) of the finite unitary groups into Harish-Chandra series is investigated. We formulate a series of conjectures relating this distribution with the crystal graph of an integrable module for a certain quantum group. Evidence for our conjectures is presented, as well as proofs for some of their consequences for the crystal graphs involved. In the course of our work we also generalize Harish-Chandra theory for some of the finite classical groups, and we introduce their Harish-Chandra branching graphs.

Let $\Irr(W)$ be the set of irreducible representations of a finite Weyl group $W$. Following an idea from Spaltenstein, Geck has recently introduced a preorder $\leq_L$ on $\Irr(W)$ in connection with the notion of Lusztig families. In a later paper with Iancu, they have shown that in type $B$ (in the asymptotic case and in the equal parameter case) this order coincides with the order on Lusztig symbols as defined by Geck and the second author in \cite{GJ}. In this paper, we show that this caracterisation extends to the so-called integer case, that is when the ratio of the parameters is an integer., Comment: 23 pages, 11 figures

The aim of this note is to study a generalization of theorems by James and Fayers on the modular representations of the symmetric group and its Hecke algebra to the case of the complex reflection groups of type $G(l,1,n)$ and the associated Ariki-Koike algebra., Comment: 13 pages

In this note, we study a factorization result for graded decomposition maps associated with the specializations of graded algebras. We obtain results previously known only in the ungraded setting., Comment: 9 pages

We give a simple characterization of the highest weight vertices in the crystal graph of the level l Fock spaces. This characterization is based on the notion of totally periodic symbols viewed as affine analogues of reverse lattice words classically used in the decomposition of tensor products of fundamental $\mathfrak{sl}_{n}$-modules. This yields a combinatorial decomposition of the Fock spaces in their irreducible components and the branching law for the restriction of the irreducible highest weight $\mathfrak{sl}_{\infty}$-modules to $\hat{\mathfrak{sl}_{e}}$., Comment: 18 pages

We study the Schur elements associated to the simple modules of the Ariki-Koike algebra. We first give a cancellation-free formula for them so that their factors can be easily read and programmed. We then study direct applications of this result. We also complete the determination of the canonical basic sets for cyclotomic Hecke algebras of type $G(l,p,n)$ in characteristic 0., Comment: The paper contains the results of arXiv:1101.1465

For the affine Hecke algebra of type A at roots of unity, we make explicit the correspondence between geometrically constructed simple modules and combinatorially constructed simple modules and prove the modular branching rule. The latter generalizes work by Vazirani., Comment: The paper has been withdrawn, see arXiv:0808.3915 for the last version.

We study the parametrizations of simple modules provided by the theory of basic sets for all finite Weyl groups. In the case of type B, we show the existence of basic sets for the matrices of constructible representations. Then we study bijections between the various basic sets and show that they are controlled by the matrices of the constructible representations.

We study the Schur elements and the a-function for cyclotomic Hecke algebras. As a consequence, we show the existence of canonical basic sets, as defined by Geck-Rouquier, for certain complex reflection groups. This includes the case of finite Weyl groups for all choices of parameters (in characteristic 0).

The level l Fock space admits canonical bases G_e and G_\infty. They correspond to U_{v}(hat{sl}_{e}) and U_{v}(sl_{\infty})-module structures. We establish that the transition matrices relating these two bases are unitriangular with coefficients in N[v]. Restriction to the highest weight modules generated by the empty l-partition then gives a natural quantization of a theorem by Geck and Rouquier on the factorization of decomposition matrices which are associated to Ariki-Koike algebras., Comment: The last version generalizes and proves the main conjecture of the previous one. Final version

We first describe how the Kashiwara involution on crystals of affine type $A$ is encoded by the combinatorics of aperiodic multisegments. This yields a simple relation between this involution and the Zelevinsky involution on the set of simple modules for the affine Hecke algebras. We then give efficient procedures for computing these involutions. Remarkably, these procedures do not use the underlying crystal structure. They also permit to match explicitly the Ginzburg and Ariki parametrizations of the simple modules associated to affine and cyclotomic Hecke algebras, respectively ., Comment: 24 pages, to appear in Pacific J. Math

For the affine Hecke algebra of type A at roots of unity, we make explicit the correspondence between geometrically constructed simple modules and combinatorially constructed simple modules and prove the modular branching rule. The latter generalizes work by Vazirani., Comment: 42 pages, last version of arXiv:1003.4838 and arXiv:0808.3915.

This note is concerned with a natural generalization of the Mullineux involution for Ariki-Koike algebras. Using a result of Fayers together with previous results by the authors, we give an efficient algorithm for computing this generalized Mullineux involution. Our algorithm notably does not involve the determination of paths in affine crystals., Comment: 17 pages

The aim of this paper is to gather and (try to) unify several approaches for the modular representation theory of Hecke algebras of type $B$. We attempt to explain the connections between Geck's cellular structures (coming from Kazhdan-Lusztig theory with unequal parameters) and Ariki's Theorem on the canonical basis of the Fock spaces.

We study the crystal graphs of irreducible $U_{v}(\hat{sl}}_{e})$-modules of higher level l. Generalizing works of the first author, we obtain a simple description of the bijections between the classes of multipartitions which naturally label these graphs: the Uglov multipartitions. This is achieved by expliciting an embedding of the $U_{v}(\hat{sl}}_{e})$-crystals of level l into $U_{v}(\hat{sl}_{\infty})$-crystals associated to highest weight modules., Comment: the revised version correct minor errors

We prove a conjecture by Dipper, James and Murphy that a bipartition is restricted if and only if it is Kleshchev. Hence the restricted bipartitions naturally label the crystal graphs of level two irreducible integrable $\mathcal{U}_v({\hat{\mathfrak{sl}}_e})$-modules and the simple modules of Hecke algebras of type $B_n$., Comment: The revised version corrects minor points, the proof of lemma 3.3 has been improved

We give a simple description of the natural bijection between the set of FLOTW bipartitions and the set of Uglov bipartitions (which generalizes the set of Kleshchev bipartitions). These bipartitions, which label the crystal graphs of irreducible $\mathcal{U}\_v({\hat{\mathfrak{sl}}\_e})$-modules of level two, naturally appear in the context of the modular representation theory of Hecke algebras of type $B\_n$.

More than 10 years ago, Dipper, James and Murphy developped the theory of Specht modules for Hecke algebras of type $B\_n$. More recently, using Lusztig's a-function, Geck and Rouquier showed how to obtain parametrisations of the irreducible representations of Hecke algebras (of any finite type) in terms of so-called canonical basic sets. For certain values of the parameters in type $B\_n$, combinatorial descriptions of these basic sets were found by Jacon, based on work of Ariki and Foda-Leclerc-Okado-Thibon-Welsh. Here, we consider the canonical basic sets for all the remaining choices of the parameters., Comment: 23 pages, to appear in J. Algebra, minor changes

We show that the different labelings of the crystal graph for irreducible highest weight $\mathcal{U}\_q (\hat{\mathfrak{sl}}\_e)$-modules lead to different labelings of the simple modules for non semisimple Ariki-Koike algebras by using Lusztig $a$-values.

This paper is a survey on the representation theory of Hecke algebras, Ariki-Koike algebras and connections with quantum group.

We give an explicit description of the "canonical basic set'' for all Iwahori-Hecke algebras of finite Weyl groups in "good'' characteristic. We obtain a complete classification of simple modules for this type of algebras.

We give a classification of the simple modules for the cyclotomic Hecke algebras over $\mathbb{C}$ in the modular case. We use the unitriangular shape of the decomposition matrices of Ariki-Koike algebras and Clifford theory.

We give a purely combinatorial algorithm for the computation of the decomposition matrices for Ariki-Koike algebras when the parameters are powers of the same root of unity. It generalizes the LLT algorithm.

Following ideas of Geck and Rouquier, we show that there exists a ``canonical basic set'' of Specht modules in bijection with the simple modules of Ariki-Koike algebras at roots of unity. Moreover, we determine the parametrization of this set and we give the consequences of these results on the representation theory of Ariki-Koike algebras., Comment: 37 pages