Hostname: page-component-5b777bbd6c-gcwzt Total loading time: 0 Render date: 2025-06-19T14:16:42.291Z Has data issue: false hasContentIssue false
  • English
  • Français

Character and class parameters from entries of character tables of symmetric groups

Part of: Representation theory of groups

Published online by Cambridge University Press:  05 June 2025

ALEXANDER ROSSI MILLER
ALEXANDER ROSSI MILLER*
Affiliation:
Center for Communications Research, Princeton e-mail: a.miller@ccr-princeton.org
Get access Rights & Permissions [Opens in a new window]

Abstract

If all of the entries of a large $S_n$ character table are covered up and you are allowed to uncover one entry at a time, then how can you quickly identify all of the indexing characters and conjugacy classes? We present a fast algorithmic solution that works even when n is so large that almost none of the entries of the character table can be computed. The fraction of the character table that needs to be uncovered is $O( n^2 \exp({-}2\pi\sqrt{n/6}))$, and for many of these entries we are only interested in whether the entry is zero.

MSC classification

Primary: 20C30: Representations of finite symmetric groups
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 9
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Chow, T. Y. and Paulhus, J.. Algorithmically distinguishing irreducible characters of the symmetric group. Electron. J. Combin. 28 (2) (2021), paper 2.5, 27 pp.10.37236/9753CrossRefGoogle Scholar
Frobenius, G.. Über die Charaktere der symmetrischen Gruppe. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin (1900), 516–534.Google Scholar
James, G. and Kerber, A.. The Representation Theory of the Symmetric Group . Encyclopedia Math. Appl. vol. 16 (Addison–Wesley, Reading, MA, 1981).Google Scholar
Macdonald, I. G.. Symmetric Functions and Hall Polynomials, 2nd ed. (Oxford University Press, 1995).10.1093/oso/9780198534891.001.0001CrossRefGoogle Scholar
Miller, A. R.. The probability that a character value is zero for the symmetric group. Math. Z. 277 (2014), 1011–1015.10.1007/s00209-014-1290-xCrossRefGoogle Scholar
Miller, A. R.. On parity and characters of symmetric groups. J. Combin. Theory Ser. A 162 (2019), 231–240.10.1016/j.jcta.2018.11.001CrossRefGoogle Scholar
Miller, A. R.. Covering numbers for characters of symmetric groups. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 26 (2025), 517–524.Google Scholar
Miller, A. R.. On Foulkes characters. Math. Ann. 381 (2021), 1589–1614.10.1007/s00208-021-02197-4CrossRefGoogle Scholar
Miller, A. R. and Scheinerman, D.. Large-scale Monte Carlo simulations for zeros in character tables of symmetric groups. Math. Comp. 94 (2025), 505–515.10.1090/mcom/3964CrossRefGoogle Scholar
Peluse, S. and Soundararajan, K.. Almost all entries in the character table of the symmetric group are multiples of any given prime. J. Reine Angew. Math. 786 (2022), 45–53.10.1515/crelle-2022-0004CrossRefGoogle Scholar
Wildon, M.. Labelling the character tables of symmetric and alternating groups. Q. J. Math. 59 (2008), 123–135.10.1093/qmath/ham024CrossRefGoogle Scholar