Search

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.

We study a natural generalization of the notion of cores for $l$-partitions attached with a multicharge ${\bf s}\in \mathbb{Z}^l$: the $(e,{\bf s})$-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., DOCUMENTA MATHEMATICA, Vol 26 (2021), p. 103-124, 1431-0643

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.

International audience; 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 Schützenberger. 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.

International audience

International audience; 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.

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., 23 pages, 11 figures

Let $W$ be a finite Weyl group and let $H$ be the associated Hecke algebra defined over the ring $A:=Z[v,v^(-1)]$ where $v$ is an indeterminate. Let $K$ be the field of fractions of $A$ and let $\theta$ be a specialisation in a field $L$. We assume that the characteristic of $L$ is ``good''. Then, by using Lusztig's $a$-function, M.Geck and R.Rouquier have defined a ``canonical basic set'' $B$ which leads to the determination of the set $\Irr(H_L)$. The aim of this work is to find explicitly this set for all $W$ and for all $\theta$ and to extend these results to the case of Ariki-Koike algebras. As consequences, we obtain an algorithm for the computation of the decomposition matrices for Ariki-Koike algebras and a characterisation of the simple modules for some cyclotomic Hecke algebras of type $G(l,l,n)$., Soit $W$ un groupe de Weyl fini et soit $H$ l'algèbre de Hecke correspondante, définie sur l'anneau $A:=Z[v,v^(-1)]$ où $v$ est une indéterminée. Soit $K$ le corps des fractions de $A$ et soit $\theta$ une spécialisation dans un corps $L$ de ``bonne'' caractéristique. Dans une série d'articles récents, M.Geck et R.Rouquier ont présenté une méthode pour déterminer l'ensemble des $H_L$-modules simples $\Irr(H_L)$. Celle-ci consiste à construire un ``ensemble basique canonique'' $B$ contenu dans $\Irr(H_K)$ défini grace à la $a$-fonction de Lusztig et en bijection avec $\Irr(H_L)$. Le but de ce travail est de déterminer explicitement $B$ pour tout groupe de Weyl et pour toute spécialisation puis d'étendre la méthode ci-dessus aux algèbres de Ariki-Koike. Comme conséquences, nous obtenons un algorithme pour le calcul des matrices de décompositions des algèbres de Ariki-Koike et une caractérisation des modules simples pour certaines algèbres cyclotomiques de type $G(l,l,n)$.

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 $\hat{\mathfrak{sl}_{e}}$., 18 pages

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

This paper is an expository paper on the representation theory of the symmetric group and its Hecke algebra in arbitrary characteristic. We study both the semisimple and the non semisimple case and give an introduction to some recent results on this theory.

On peut associer à tout groupe de réflexions complexes, son algèbre de Hecke H(W). Celle-ci peut etre vue comme une déformation de l'algèbre du groupe W. La théorie d'Ariki-Lascoux-Leclerc-Thibon a permis de montrer que les représentations de ces algèbres sont dans certains cas intimement reliées à des objets remarquables provenant de la théorie des groupes quantiques en type A affine (comme les cristaux ou les bases canoniques de Kashiwara-Lusztig). Le principal objectif de ce mémoire est d'étudier puis d'étendre les liens unissant ces deux théories. Nous obtenons entre autres des paramétrisations des modules simples pour H(W) grace à l'étude des cristaux du groupe quantique, calculons les matrices de décompositions associées ou encore étudions certaines involutions remarquables de H(W). Des résultats concernant la théorie des représentations des algèbres de Hecke affines de type A sont également présent\és (règle de branchement modulaire, calcul de l'involution de Zelevinsky etc.)

On peut associer à tout groupe de réflexions complexes, son algèbre de Hecke H(W). Celle-ci peut etre vue comme une déformation de l'algèbre du groupe W. La théorie d'Ariki-Lascoux-Leclerc-Thibon a permis de montrer que les représentations de ces algèbres sont dans certains cas intimement reliées à des objets remarquables provenant de la théorie des groupes quantiques en type A affine (comme les cristaux ou les bases canoniques de Kashiwara-Lusztig). Le principal objectif de ce mémoire est d'étudier puis d'étendre les liens unissant ces deux théories. Nous obtenons entre autres des paramétrisations des modules simples pour H(W) grace à l'étude des cristaux du groupe quantique, calculons les matrices de décompositions associées ou encore étudions certaines involutions remarquables de H(W). Des résultats concernant la théorie des représentations des algèbres de Hecke affines de type A sont également présent\és (règle de branchement modulaire, calcul de l'involution de Zelevinsky etc.)

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., the revised version correct minor errors

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

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

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

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

International audience; 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.

8 pages; Dans cette note, nous déterminons explicitement les cellules de Kazhdan et Lusztig contenant les éléments de longueur maximale dans les sous-groupes paraboliques du groupe symétrique associés à une partition de 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

International audience; 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.

International audience; 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.

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

Let $W$ be a finite Weyl group and let $H$ be the associated Hecke algebra defined over the ring $A:=Z[v,v^(-1)]$ where $v$ is an indeterminate. Let $K$ be the field of fractions of $A$ and let $\theta$ be a specialisation in a field $L$. We assume that the characteristic of $L$ is ``good''. Then, by using Lusztig's $a$-function, M.Geck and R.Rouquier have defined a ``canonical basic set'' $B$ which leads to the determination of the set $\Irr(H_L)$. The aim of this work is to find explicitly this set for all $W$ and for all $\theta$ and to extend these results to the case of Ariki-Koike algebras. As consequences, we obtain an algorithm for the computation of the decomposition matrices for Ariki-Koike algebras and a characterisation of the simple modules for some cyclotomic Hecke algebras of type $G(l,l,n)$.; Soit $W$ un groupe de Weyl fini et soit $H$ l'algèbre de Hecke correspondante, définie sur l'anneau $A:=Z[v,v^(-1)]$ où $v$ est une indéterminée. Soit $K$ le corps des fractions de $A$ et soit $\theta$ une spécialisation dans un corps $L$ de ``bonne'' caractéristique. Dans une série d'articles récents, M.Geck et R.Rouquier ont présenté une méthode pour déterminer l'ensemble des $H_L$-modules simples $\Irr(H_L)$. Celle-ci consiste à construire un ``ensemble basique canonique'' $B$ contenu dans $\Irr(H_K)$ défini grace à la $a$-fonction de Lusztig et en bijection avec $\Irr(H_L)$. Le but de ce travail est de déterminer explicitement $B$ pour tout groupe de Weyl et pour toute spécialisation puis d'étendre la méthode ci-dessus aux algèbres de Ariki-Koike. Comme conséquences, nous obtenons un algorithme pour le calcul des matrices de décompositions des algèbres de Ariki-Koike et une caractérisation des modules simples pour certaines algèbres cyclotomiques de type $G(l,l,n)$.