Abstract
Let i=1+q+⋅⋅⋅+q i−1. For certain sequences (r 1,…,r l ) of positive integers, we show that in the Hecke algebra ℋ n (q) of the symmetric group \(\mathfrak{S}_{n}\) , the product \((1+\boldsymbol{r}_{\boldsymbol{1}}T_{r_{1}})\cdots (1+\boldsymbol{r}_{\boldsymbol{l}}T_{r_{l}})\) has a simple explicit expansion in terms of the standard basis {T w }. An interpretation is given in terms of random walks on \(\mathfrak{S}_{n}\) .
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Cherednik, I.V.: Special bases of irreducible representations of a degenerate affine Hecke algebra. Funct. Analysis Appl. 20, 76–79 (1986). Translated from Funktsional. Anal. i Prilozhen. 20, 87–88 (1986)
Cherednik, I.V.: A new interpretation of Gel’fand Tzetlin bases. Duke Math. J. 54, 563–577 (1987)
Diaconis, P.: Group Representations in Probability and Statistics. Lecture Notes–Monograph Series, vol. 11. Institute of Mathematical Statistics, Hayward (1988)
Diaconis, P., Ram, A.: Analysis of systematic scan Metropolis algorithms using Iwahori-Hecke algebra techniques. Michigan Math. J. 48, 157–190 (2000)
Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990)
Jucys, A.: On the Young operators of symmetric groups. Litovsk. Fiz. Sb. 6, 163–180 (1966)
Author information
Authors and Affiliations
Corresponding author
Additional information
Rosena Du is partially supported by the National Science Foundation of China under Grant No. 10801053 and No. 10671074. Research carried out when she was a Visiting Scholar at M.I.T. during the 2007–2008 academic year. Richard Stanley’s contribution is based upon work supported by the National Science Foundation under Grant No. DMS-0604423. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect those of the National Science Foundation.
Rights and permissions
About this article
Cite this article
Du, R.R.X., Stanley, R.P. Some Hecke algebra products and corresponding random walks. J Algebr Comb 31, 159–168 (2010). https://doi.org/10.1007/s10801-009-0193-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10801-009-0193-0