142 research outputs found
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 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 Fock
spaces representations of . 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 . 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) âKn, 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 HoefsmitâČs [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 BrouĂ© 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 BrouĂ© 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
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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 -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 introduced by F. Aicardi and J.
Juyumaya as an abstraction of the Yokonuma-Hecke algebra. We construct a tensor
space representation for and show that this is faithful. We use
it to give a basis for 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 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
gives rise to graded dual Hopf algebras then we must
have where .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 is applied to type 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 .Comment: to appear in Algebras and Representation theor
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