# On averages of randomized class functions on the symmetric groups and their asymptotics

Paul-Olivier Dehaye^{[1]}; Dirk Zeindler^{[2]}

- [1] Institut für Mathematik Universität Zürich Winterthurerstrasse 190 CH-8057 Zürich
- [2] Universität Bielefeld SFB 701 Postfach: 100 131 33501 Bielefeld Deutschland

Annales de l’institut Fourier (2013)

- Volume: 63, Issue: 4, page 1227-1262
- ISSN: 0373-0956

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topDehaye, Paul-Olivier, and Zeindler, Dirk. "On averages of randomized class functions on the symmetric groups and their asymptotics." Annales de l’institut Fourier 63.4 (2013): 1227-1262. <http://eudml.org/doc/275640>.

@article{Dehaye2013,

abstract = {The second author had previously obtained explicit generating functions for moments of characteristic polynomials of permutation matrices ($n$ points). In this paper, we generalize many aspects of this situation. We introduce random shifts of the eigenvalues of the permutation matrices, in two different ways: independently or not for each subset of eigenvalues associated to the same cycle. We also consider vastly more general functions than the characteristic polynomial of a permutation matrix, by first finding an equivalent definition in terms of cycle-type of the permutation. We consider other groups than the symmetric group, for instance the alternating group and other Weyl groups. Finally, we compute some asymptotics results when $n$ tends to infinity. This last result requires additional ideas: it exploits properties of the Feller coupling, which gives asymptotics for the lengths of cycles in permutations of many points.},

affiliation = {Institut für Mathematik Universität Zürich Winterthurerstrasse 190 CH-8057 Zürich; Universität Bielefeld SFB 701 Postfach: 100 131 33501 Bielefeld Deutschland},

author = {Dehaye, Paul-Olivier, Zeindler, Dirk},

journal = {Annales de l’institut Fourier},

keywords = {symmetric group; characteristic polynomial; associated class functions; generating functions; Feller coupling; asymptotics of moments},

language = {eng},

number = {4},

pages = {1227-1262},

publisher = {Association des Annales de l’institut Fourier},

title = {On averages of randomized class functions on the symmetric groups and their asymptotics},

url = {http://eudml.org/doc/275640},

volume = {63},

year = {2013},

}

TY - JOUR

AU - Dehaye, Paul-Olivier

AU - Zeindler, Dirk

TI - On averages of randomized class functions on the symmetric groups and their asymptotics

JO - Annales de l’institut Fourier

PY - 2013

PB - Association des Annales de l’institut Fourier

VL - 63

IS - 4

SP - 1227

EP - 1262

AB - The second author had previously obtained explicit generating functions for moments of characteristic polynomials of permutation matrices ($n$ points). In this paper, we generalize many aspects of this situation. We introduce random shifts of the eigenvalues of the permutation matrices, in two different ways: independently or not for each subset of eigenvalues associated to the same cycle. We also consider vastly more general functions than the characteristic polynomial of a permutation matrix, by first finding an equivalent definition in terms of cycle-type of the permutation. We consider other groups than the symmetric group, for instance the alternating group and other Weyl groups. Finally, we compute some asymptotics results when $n$ tends to infinity. This last result requires additional ideas: it exploits properties of the Feller coupling, which gives asymptotics for the lengths of cycles in permutations of many points.

LA - eng

KW - symmetric group; characteristic polynomial; associated class functions; generating functions; Feller coupling; asymptotics of moments

UR - http://eudml.org/doc/275640

ER -

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