Cyclotomic Shuffles

  title={Cyclotomic Shuffles},
  author={O. Ogievetsky and Varvara Petrova},
  journal={Physics of Particles and Nuclei},
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… Expand

Figures from this paper

Progressive Randomization of a Deck of Playing Cards: Experimental Tests and Statistical Analysis of the Riffle Shuffle
The question of how many shuffles are required to randomize an initially ordered deck of cards is a problem that has fascinated mathematicians, scientists, and the general public. The two principalExpand


Braids, shuffles and symmetrizers*
Multiplicative analogues of the shuffle elements of the braid group rings are introduced; in local representations they give rise to certain graded associative algebras (b-shuffle algebras). For theExpand
Induced representations and traces for chains of affine and cyclotomic Hecke algebras
Abstract Properties of relative traces and symmetrizing forms on chains of cyclotomic and affine Hecke algebras are studied. The study relies on the use of bases of these algebras which generalize aExpand
An inductive approach to representations of complex reflection groups G(m, 1, n)
We propose an inductive approach to the representation theory of the chain of complex reflection groups G(m, 1, n). We obtain the Jucys-Murphy elements of G(m, 1, n) from the Jucys-Murphy elements ofExpand
A Q-analogue of an Identity of N.wallach
1. Let n be an integer ≥ 2. For i ∈ [1, n − 1] let s i be the transposition (i, i + 1) in the group S n of permutations of [1, n]. (Given two integers a, b we denote by [a, b] the set of all integersExpand
A q-analogue of an identity of N. Wallach
N. Wallach has considered an element of the group algebra of the symmetric group S n which is the sum of an n-cycle, an (n − 1)-cycle, . . . , a 2-cycle and the identity. He showed thatExpand
Qsym over Sym is free
An algorithm is presented that recursively constructs a Λn-module basis for 2Jn thereby proving one of the Bergeron-Reutenauer conjectures and implying that the quotient of 2JN by the ideal generated by the elementary symmetric functions has dimension n!. Expand
Analysis of Top To Random Shuffles
A deck of n cards is shuffled by repeatedly taking off the top m cards and inserting them in random positions to give the asymptotics of the approach to stationarity: for m fixed and n large, it takes shuffles to get close to random. Expand
A practical method for enumerating cosets of a finite abstract group
An important problem in finite-group theory is the determination of an abstract definition for a given group , that is, a set of relations between k generating operations S 1 , …., S k of , such thatExpand
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–MurphyExpand
Combinatorial Aspects of Multiple Zeta Values
This work proves a longstanding conjecture of Don Zagier about MZVs with certain repeated arguments and presents extensive computational evidence supporting an infinite family of conjectured MZV identities that simultaneously generalize the ZagIER identity. Expand