Cyclotomic Shuffles

摘要

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.