<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume9 Issue1" %>
Expected Reflection Distance in G(r, 1, n) After a Fixed Number of Reflections
Niklas Eriksen1, and Axel Hultman2
1Department of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden
2Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, 35032 Marburg Germany
Annals of Combinatorics 9 (1) p.21-33 March, 2005
AMS Subject Classification: 05C12, 82B41, 51F15
Extending to r > 1 a formula of the authors, we compute the expected reflection distance of a product of t random reflections in the complex reflection group G(r, 1, n). The result relies on an explicit decomposition of the reflection distance function into irreducible G(r, 1, n)- characters and on the eigenvalues of certain adjacency matrices.
Keywords: complex reflection groups, reflection distances, random walks


1. S. Ariki and K. Koike, A Hecke algebra of and construction of its irreducible representations, Adv. Math. 106 (1994) 216--243.

2. P. Diaconis, Group Representations in Probability and Statistics, IMS, Hayward, CA, 1988.

3. M. Dyer, On the "Bruhat graph" of a Coxeter system, Compositio Math. 78 (1991) 185--191.

4. N. Eriksen and A. Hultman, Estimating the expected reversal distance after a fixed number of reversals, Adv. Appl. Math. 32 (2004) 439--453.

5. J.A. Fill and C.H. Schoolfield Jr., Mixing times for Markov chains on wreath products and related homogeneous spaces, Electron. J. Probab. 6 (2001) paper 11.

6. N. Ito, The spectrum of a conjugacy class graph of a finite group, Math. J. Okayama Univ. 26 (1984) 1--10.

7. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Second Edition, Oxford University Press, Oxford, 1995.

8. C.H. Schoolfield Jr., Random walks on wreath products of groups, J. Theoret. Probab. 15 (2002) 667--693.

9. G.C. Shephard, Regular complex polytopes, Proc. London Math. Soc. 2 (1952) 82--97.

10. R.P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, New York/Cambridge, 1999.