135 research outputs found
Cyclotomic shuffles
Analogues of 1-shuffle elements for complex reflection groups of type
are introduced. A geometric interpretation for in terms
of rotational permutations of polygonal cards is given. We compute the
eigenvalues, and their multiplicities, of the 1-shuffle element in the algebra
of the group . Considering shuffling as a random walk on the group
, we estimate the rate of convergence to randomness of the
corresponding Markov chain. We report on the spectrum of the 1-shuffle analogue
in the cyclotomic Hecke algebra for and small
BRST operator for quantum Lie algebras and differential calculus on quantum groups
For a Hopf algebra A, we define the structures of differential complexes on
two dual exterior Hopf algebras: 1) an exterior extension of A and 2) an
exterior extension of the dual algebra A^*. The Heisenberg double of these two
exterior Hopf algebras defines the differential algebra for the Cartan
differential calculus on A. The first differential complex is an analog of the
de Rham complex. In the situation when A^* is a universal enveloping of a Lie
(super)algebra the second complex coincides with the standard complex. The
differential is realized as an (anti)commutator with a BRST- operator Q. A
recurrent relation which defines uniquely the operator Q is given. The BRST and
anti-BRST operators are constructed explicitly and the Hodge decomposition
theorem is formulated for the case of the quantum Lie algebra U_q(gl(N)).Comment: 20 pages, LaTeX, Lecture given at the Workshop on "Classical and
Quantum Integrable Systems", 8 - 11 January, Protvino, Russia; corrected some
typo
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
Baxterized Solutions of Reflection Equation and Integrable Chain Models
Non-polynomial Baxterized solutions of reflection equations associated with
affine Hecke and affine Birman-Murakami-Wenzl algebras are found. Relations to
integrable spin chain models with nontrivial boundary conditions are discussed.Comment: LaTeX, 18 pages, typos fixed, journal-ref adde
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
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