Abstract
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 \(n\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
S. Ariki and K. Koike, “A Hecke algebra of \(\left( {{\mathbb{Z} \mathord{\left/ {\vphantom {\mathbb{Z} {r\mathbb{Z}}}} \right. \kern-0em} {r\mathbb{Z}}}} \right)\wr {{S}_{n}}\) and construction of its irreducible representations,” Adv. Math. 106 (2), 216 (1994).
J. M. Borwein, D. M. Bradley, D. J. Broadhurst, and P. Lisonek, “Combinatorial aspects of multiple zeta values,” Electr. J. Combinatorics 5 (1), R38 (1998).
D. Bowman and D. Bradley, “Multiple polylogarithms: A brief survey,” Contemp. Math. 291, 71 (2001).
H. Coxeter and J. Todd, “A practical method for enumerating cosets of a finite abstract group,” Proc. Edinburgh Math. Soc. 5, 26 (1936).
P. Diaconis, J. A. Fill, and J. Pitman, “Analysis of top to random shuffles,” Combinatorics, Probab. Comput. 1, 135 (1992).
A. Garsia and N. Wallach, “Qsym over Sym is free,” J. Combinatorial Theory, Ser. A 104, 217 (2003).
T. Grapperon and O. V. Ogievetsky, “Braidings of tensor spaces,” Lett. Math. Phys. 100, 17 (2012).
A. P. Isaev and O. V. Ogievetsky, “BRST operator for quantum Lie algebras: Explicit formula,” Int. J. Mod. Phys. A 19, 240 (2004).
A. P. Isaev and O. V. Ogievetsky, “Braids, shuffles and symmetrizers,” J. Phys. A: Math. Theor. 42, 1 (2009).
P. Lorek, “Speed of convergence to stationarity for stochastically monotone Markov chains,” PhD Thesis (Math. Inst. Univ. Wroclaw, 2007).
G. Lusztig, “A q-analogue of an identity of N. Wallach,” in Studies in Lie Theory, 405 (2006).
S. Mac Lane, Homology, Classics in Mathematics, Vol. 114 (Springer, 1963).
W. D. Nichols, “Bialgebras of type one,” Commun. Algebra 6, 1521 (1978).
O. V. Ogievetsky and L. Poulain d’Andecy, “On representations of cyclotomic Hecke algebras,” Mod. Phys. Lett. A 26, 795 (2011).
O. V. Ogievetsky and L. Poulain d’Andecy, “An inductive approach to representations of complex reflection groups \(G(m,1,n)\),” Theor. Math. Phys. 174, 95 (2013).
O. V. Ogievetsky and L. Poulain d’Andecy, “Induced representations and traces for chains of affine and cyclotomic Hecke algebras,” J. Geom. Phys. 87, 354 (2015).
A. Okounkov and A. Vershik, “A new approach to representation theory of symmetric groups II,” Selecta Math (New Ser.) 2, 581 (1996).
R. M. Phatarfod, “On the matrix occurring in a linear search problem,” J. Appl. Probab. 28, 336 (1991).
E. Seneta, Non-Negative Matrices and Markov Chains (Springer, 2006).
N. R. Wallach, “Lie algebra cohomology and holomorphic continuation of generalized Jacquet integrals,” Representations of Lie Groups (Kyoto, Hiroshima, 1986), vol. 14, p. 123.
ACKNOWLEDGMENTS
The work of O. O. was supported by the Program of Competitive Growth of Kazan Federal University and by the grant RFBR 17-01-00585. The work of V. P. has been carried out thanks to the support of the A*MIDEX grant (ANR-11-IDEX-0001-02) funded by the French Government Investissements d’Avenir program.
Author information
Authors and Affiliations
Corresponding author
Additional information
1The article is published in the original.
Rights and permissions
About this article
Cite this article
Ogievetsky, O., Petrova, V. Cyclotomic Shuffles. Phys. Part. Nuclei 49, 867–872 (2018). https://doi.org/10.1134/S1063779618050325
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1063779618050325