1,344 research outputs found
Cyclic generators for irreducible representations of affine Hecke algebras
We give a detailed account of a combinatorial construction, due to Cherednik,
of cyclic generators for irreducible modules of the affine Hecke algebra of the
general linear group with generic parameter q.Comment: Latex file, 24 pages, minor corrections. To appear in Journal of
Combinatorial Theory
Periodic cyclic homology of Iwahori-Hecke algebras
We determine the periodic cyclic homology of the Iwahori-Hecke algebras
\Hecke_q, for q \in \CC^* not a ``proper root of unity.'' (In this paper,
by a {\em proper root of unity} we shall mean a root of unity other than 1.)
Our method is based on a general result on periodic cyclic homology, which
states that a ``weakly spectrum preserving'' morphism of finite type algebras
induces an isomorphism in periodic cyclic homology. The concept of a weakly
spectrum preserving morphism is defined in this paper, and most of our work is
devoted to understanding this class of morphisms. Results of Kazhdan--Lusztig
and Lusztig show that, for the indicated values of , there exists a weakly
spectrum preserving morphism \phi_q : \Hecke_q \to J, to a fixed finite type
algebra . This proves that induces an isomorphism in periodic
cyclic homology and, in particular, that all algebras \Hecke_q have the same
periodic cyclic homology, for the indicated values of . The periodic cyclic
homology groups of the algebra \Hecke_1 can then be determined directly,
using results of Karoubi and Burghelea, because it is the group algebra of an
extended affine Weyl group.Comment: 24 pages, LaTe
On representations of complex reflection groups G(m,1,n)
An inductive approach to the representation theory of the chain of the
complex reflection groups G(m,1,n) is presented. We obtain the Jucys-Murphy
elements of G(m,1,n) from the Jucys--Murphy elements of the cyclotomic Hecke
algebra, and study their common spectrum using representations of a degenerate
cyclotomic affine Hecke algebra. Representations of G(m,1,n) are constructed
with the help of a new associative algebra whose underlying vector space is the
tensor product of the group ring of G(m,1,n) with a free associative algebra
generated by the standard m-tableaux.Comment: 18 page
Degenerate Affine Hecke-Clifford Algebras and Type Lie Superalgebras
We construct the finite dimensional simple integral modules for the
(degenerate) affine Hecke-Clifford algebra (AHCA). Our construction includes an
analogue of Zelevinsky's segment representations, a complete combinatorial
description of the simple calibrated modules, and a classification of the
simple integral modules. Additionally, we construct an analogue of the
Arakawa-Suzuki functor for the Lie superalgebra of type Q.Comment: 66 pages. Section 8 revise
On representations of cyclotomic Hecke algebras
An approach, based on Jucys--Murphy elements, to the representation theory of
cyclotomic Hecke algebras is developed. The maximality (in the cyclotomic Hecke
algebra) of the set of the Jucys--Murphy elements is established. A basis of
the cyclotomic Hecke algebra is suggested; this basis is used to establish the
flatness of the deformation without use of the representation theory
Generalized double affine Hecke algebras of rank 1 and quantized Del Pezzo surfaces
Let D be a simply laced Dynkin diagram of rank r whose affinization has the
shape of a star (i.e., D4,E6,E7,E8). To such a diagram one can attach a group G
whose generators correspond to the legs of the affinization, have orders equal
to the leg lengths plus 1, and the product of the generators is 1. The group G
is then a 2-dimensional crystallographic group: G=Z_l\ltimes Z^2, where l is
2,3,4, and 6, respectively. In this paper, we define a flat deformation H(t,q)
of the group algebra C[G] of this group, by replacing the relations saying that
the generators have prescribed orders by their deformations, saying that the
generators satisfy monic polynomial equations of these orders with arbitrary
roots (which are deformation parameters). The algebra H(t,q) for D4 is the
Cherednik algebra of type C^\check C_1, which was studied by Noumi, Sahi, and
Stokman, and controls Askey-Wilson polynomials. We prove that H(t,q) is the
universal deformation of the twisted group algebra of G, and that this
deformation is compatible with certain filtrations on C[G]. We also show that
if q is a root of unity, then for generic t the algebra H(t,q) is an Azumaya
algebra, and its center is the function algebra on an affine del Pezzo surface.
For generic q, the spherical subalgebra eH(t,q)e provides a quantization of
such surfaces. We also discuss connections of H(t,q) with preprojective
algebras and Painlev\'e VI.Comment: 44 pages, latex; in the new version there is a new appendix by W.
Crawley-Boevey and P.Sha
A category for the adjoint representation
We construct an abelian category C and exact functors in C which on the
Grothendieck group descend to the action of a simply-laced quantum group in its
adjoint representation. The braid group action in the adjoint representation
lifts to an action in the derived category of C. The category C is the direct
sum of a semisimple category and the category of modules over a certain algebra
A, associated to a Dynkin diagram. In the second half of the paper we show how
these algebras appear in the modular representation theory and in the McKay
correspondence and explore their relationship with root systems.Comment: latex file + 4 eps files with figures; several mistakes found in the
original version were corrected, in particular propositions 16-18 required
additional assumption of binary
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