Cyclotomic shuffles

Analogues of 1-shuffle elements for complex reflection groups of type $G(m,1,n)$ are introduced. A geometric interpretation for $G(m,1,n)$ 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 $G(m,1,n)$. Considering shuffling as a random walk on the group $G(m,1,n)$, 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 $H(m,1,n)$ for $m=2$ and small $n$

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