Schur elements for the Ariki-Koike algebra and applications

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:1101.146

Crystal isomorphisms in Fock spaces and Schensted correspondence in affine type A

We are interested in the structure of the crystal graph of level $l$ Fock spaces representations of $\mathcal{U}_q (\widehat{\mathfrak{sl}_e})$. Since the work of Shan [26], we know that this graph encodes the modular branching rule for a corresponding cyclotomic rational Cherednik algebra. Besides, it appears to be closely related to the Harish-Chandra branching graph for the appropriate finite unitary group, according to [8]. In this paper, we make explicit a particular isomorphism between connected components of the crystal graphs of Fock spaces. This so-called "canonical" crystal isomorphism turns out to be expressible only in terms of: - Schensted's classic bumping procedure, - the cyclage isomorphism defined in [13], - a new crystal isomorphism, easy to describe, acting on cylindric multipartitions. We explain how this can be seen as an analogue of the bumping algorithm for affine type $A$. Moreover, it yields a combinatorial characterisation of the vertices of any connected component of the crystal of the Fock space

Representation Theory of a Hecke Algebra of G(r, p, n)

AbstractIn our previous paper [1], we introduced a notion of a Hecke algebra for a series of complex reflection groups (Z/rZ) ≀Kn, which is the group of n by n permutation matrices with entries running through rth roots of unity. The ordinary representation of the algebra is also studied in it. The results are direct generalizations of Hoefsmit′s [4] for Hecke algebras of classical Weyl groups. In this paper, we establish a similar theory for the remaining series of complex reflection groups G(r, p, n) (p\r). p = 1 is the case we previously considered in [1]. The construction of the Hecke algebra was soon generalized to the case p = 2 by Broué and Malle [2], motivated by a much deeper background. In [2], they conjectured that these Hecke algebras over a ring of l-adic integers are candidates for endomorphism rings of certain Deligne-Lusztig modules over the ring, which arise in modular representation theory related to algebraic groups. They also defined Hecke algebras of complex reflection groups for (not all but various kinds of) exceptional and two-dimensional cases in the same context. The first section starts with the definition of the Hecke algebra and then we prove that they actually have desirable properties. The second section is for explicit construction of irreducible representations. We call them the representations of semi-normal form. The author deeply thanks Professor Broué and Professor Malle, since he was able to learn much from them, and they hinted him to the problem

Nilpotency in type A cyclotomic quotients

We prove a conjecture made by Brundan and Kleshchev on the nilpotency degree of cyclotomic quotients of rings that categorify one-half of quantum sl(k).Comment: 19 pages, 39 eps files. v3 simplifies antigravity moves and corrects typo

Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras

We construct an explicit isomorphism between blocks of cyclotomic Hecke algebras and (sign-modified) Khovanov-Lauda algebras in type A. These isomorphisms connect the categorification conjecture of Khovanov and Lauda to Ariki's categorification theorem. The Khovanov-Lauda algebras are naturally graded, which allows us to exhibit a non-trivial Z-grading on blocks of cyclotomic Hecke algebras, including symmetric groups in positive characteristic.Comment: 32 pages; minor changes to section

Weight Vectors of the Basic A_1^(1)-Module and the Littlewood-Richardson Rule

The basic representation of \A is studied. The weight vectors are represented in terms of Schur functions. A suitable base of any weight space is given. Littlewood-Richardson rule appears in the linear relations among weight vectors.Comment: February 1995, 7pages, Using AMS-Te

Compound basis arising from the basic A1(1)A^{(1)}_{1}-module

A new basis for the polynomial ring of infinitely many variables is constructed which consists of products of Schur functions and Q-functions. The transition matrix from the natural Schur function basis is investigated.Comment: 12 page

On the Representation Theory of an Algebra of Braids and Ties

We consider the algebra ${\cal E}_n(u)$ introduced by F. Aicardi and J. Juyumaya as an abstraction of the Yokonuma-Hecke algebra. We construct a tensor space representation for ${\cal E}_n(u)$ and show that this is faithful. We use it to give a basis for ${\cal E}_n(u)$ and to classify its irreducible representations.Comment: 24 pages. Final version. To appear in Journal of Algebraic Combinatorics

Combinatorial Hopf algebras and Towers of Algebras

Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras $\bigoplus_{n\ge0}A_n$ can be endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap, and independently Lam and Shimozono constructed dual graded graphs from primitive elements in Hopf algebras. In this paper we apply the composition of these constructions to towers of algebras. We show that if a tower $\bigoplus_{n\ge0}A_n$ gives rise to graded dual Hopf algebras then we must have $\dim(A_n)=r^nn!$ where $r = \dim(A_1)$.Comment: 7 page

On representation theory of affine Hecke algebras of type B

Ariki's and Grojnowski's approach to the representation theory of affine Hecke algebras of type $A$ is applied to type $B$ with unequal parameters to obtain -- under certain restrictions on the eigenvalues of the lattice operators -- analogous multiplicity-one results and a classification of irreducibles with partial branching rules as in type $A$.Comment: to appear in Algebras and Representation theor