Group characters, permutation actions and sharpness

Kenneth Johnson, Eirini Poimenidou

Research output: Contribution to journalArticle

Abstract

We extend the work which has appeared in papers on sharp characters and originated with Blichfeldt and Maillet to the Burnside ring of a finite group G. We show that the polynomial whose zeros are the set of marks of non-identity subgroups on a faithful G-set X evaluated at X is an integral multiple of the regular G-set, and deduce a result about the size of a base of X. Further consequences for ordinary group characters are obtained by re-examining Blichfeldt's work and we provide alternative definitions of sharpness. Conjectures are given related to the set of values of a permutation character, and it is proved that for a faithful transitive G-set X certain polynomials (in the Burnside ring) evaluated at X necessarily give G-sets.

Original languageEnglish (US)
Pages (from-to)173-182
Number of pages10
JournalEuropean Journal of Combinatorics
Volume24
Issue number2
DOIs
StatePublished - Feb 1 2003

Fingerprint

Sharpness
Permutation
Burnside Ring
Faithful
Polynomial Zeros
Multiple integral
Character
Deduce
Finite Group
Subgroup
Polynomial
Alternatives

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics

Cite this

Johnson, Kenneth ; Poimenidou, Eirini. / Group characters, permutation actions and sharpness. In: European Journal of Combinatorics. 2003 ; Vol. 24, No. 2. pp. 173-182.
@article{143ef48c253d40878c35a2cebb70bd3f,
title = "Group characters, permutation actions and sharpness",
abstract = "We extend the work which has appeared in papers on sharp characters and originated with Blichfeldt and Maillet to the Burnside ring of a finite group G. We show that the polynomial whose zeros are the set of marks of non-identity subgroups on a faithful G-set X evaluated at X is an integral multiple of the regular G-set, and deduce a result about the size of a base of X. Further consequences for ordinary group characters are obtained by re-examining Blichfeldt's work and we provide alternative definitions of sharpness. Conjectures are given related to the set of values of a permutation character, and it is proved that for a faithful transitive G-set X certain polynomials (in the Burnside ring) evaluated at X necessarily give G-sets.",
author = "Kenneth Johnson and Eirini Poimenidou",
year = "2003",
month = "2",
day = "1",
doi = "10.1016/S0195-6698(02)00146-4",
language = "English (US)",
volume = "24",
pages = "173--182",
journal = "European Journal of Combinatorics",
issn = "0195-6698",
publisher = "Academic Press Inc.",
number = "2",

}

Group characters, permutation actions and sharpness. / Johnson, Kenneth; Poimenidou, Eirini.

In: European Journal of Combinatorics, Vol. 24, No. 2, 01.02.2003, p. 173-182.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Group characters, permutation actions and sharpness

AU - Johnson, Kenneth

AU - Poimenidou, Eirini

PY - 2003/2/1

Y1 - 2003/2/1

N2 - We extend the work which has appeared in papers on sharp characters and originated with Blichfeldt and Maillet to the Burnside ring of a finite group G. We show that the polynomial whose zeros are the set of marks of non-identity subgroups on a faithful G-set X evaluated at X is an integral multiple of the regular G-set, and deduce a result about the size of a base of X. Further consequences for ordinary group characters are obtained by re-examining Blichfeldt's work and we provide alternative definitions of sharpness. Conjectures are given related to the set of values of a permutation character, and it is proved that for a faithful transitive G-set X certain polynomials (in the Burnside ring) evaluated at X necessarily give G-sets.

AB - We extend the work which has appeared in papers on sharp characters and originated with Blichfeldt and Maillet to the Burnside ring of a finite group G. We show that the polynomial whose zeros are the set of marks of non-identity subgroups on a faithful G-set X evaluated at X is an integral multiple of the regular G-set, and deduce a result about the size of a base of X. Further consequences for ordinary group characters are obtained by re-examining Blichfeldt's work and we provide alternative definitions of sharpness. Conjectures are given related to the set of values of a permutation character, and it is proved that for a faithful transitive G-set X certain polynomials (in the Burnside ring) evaluated at X necessarily give G-sets.

UR - http://www.scopus.com/inward/record.url?scp=0037223229&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037223229&partnerID=8YFLogxK

U2 - 10.1016/S0195-6698(02)00146-4

DO - 10.1016/S0195-6698(02)00146-4

M3 - Article

VL - 24

SP - 173

EP - 182

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

SN - 0195-6698

IS - 2

ER -