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

Abstract

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The second author had previously obtained explicit generating functions for moments of characteristic polynomials of permutation matrices ( 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 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.

How to cite

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Dehaye, 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 -

References

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