Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T14:53:17.309Z Has data issue: false hasContentIssue false
  • English
  • Français

A formal power series attached to a class function on a group and its application to the characterisation of characters

Published online by Cambridge University Press:  03 July 2013

KENNETH W. JOHNSON  and
EIRINI POIMENIDOU
KENNETH W. JOHNSON
Affiliation:
Abington College, The Pennsylvania State University, 1600 Woodland Road, Abington, PA 19001, U.S.A. e-mail: [email protected]
EIRINI POIMENIDOU
Affiliation:
Division of Natural Sciences, New College of Florida, 5800 Bay Shore Road, Sarasota, FL 34243, U.S.A. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In Frobenius' initial papers on group characters he introduced k-characters in order to give an algorithm to calculate the irreducible factors of the group determinant. We show how his work leads naturally to the construction of a formal power series for any class function on a group, which terminates if and only if the class function is a character. This is then used to obtain a criterion for a class function to be a character.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 3 , November 2013 , pp. 465 - 474
Copyright
Copyright © Cambridge Philosophical Society 2013 

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.)

References

REFERENCES

[1]Bergman, A.Hauptnorm und Struktur von Algebren. J. Reine Angew Math. 222 (1960), 160–194.Google Scholar
[2]Bergman, A.Reduziere Normen and Theorie von Algebren. Algebra. Tagung Halle 1986, Tagungsband pp. 29–57, Wiss. Beitrage Halle (Saale) 1987/33 (M48), (1987).Google Scholar
[3]Buchstaber, V. M. and Rees, E. G.The Gel'fand map and symmetric products. Selecta Math. (N.S.) 8 (2002), 523–535.CrossRefGoogle Scholar
[4]Buchstaber, V. M. and Rees, E.GRings of continuous functions, symmetric products and Frobenius Algebras. Russian Math. Surveys 59 (2004), 125–144.CrossRefGoogle Scholar
[5]Cox, D., Little, J. and O'Shea, D.Ideals, Varieties and Algorithms. 3rd Edition (Springer, 2007).CrossRefGoogle Scholar
[6]Curtis, C. W.Pioneers of Representation theory, Frobenius, Burnside, Schur and Brauer. American Math. Soc. (1999).CrossRefGoogle Scholar
[7]Dickson, L. E.An elementary exposition of Frobenius's theory of group character and group determinants. Ann. of Math. (2) 4 (1902) 285–301.CrossRefGoogle Scholar
[8]Frobenius, G.Über Gruppencharaktere, Sitzungsber. Preuss. Akad. Wiss. Berlin (1896), 985–1021. (Gesammelte Abhandlungen, (Springer-Verlag, 1968), 1–37).Google Scholar
[9]Frobenius, G.Über die Primfactoren der Gruppendeterminante, Sitzungsber. Preuss Akad. Wiss Berlin (1896), 1343-1382. (Gesammelte Abhandlungen, (Springer-Verlag, 1968), 38–77).Google Scholar
[10]Hawkins, T.New light on Frobenius' creation of the theory of group characters. Arch. History Exact Sci. 12 (1974), 217–243.CrossRefGoogle Scholar
[11]Helling, H.Eine kennzeichnung von charakteren auf gruppen und associativen algebren. Comm. Algebra 1 (1974), 491–501.CrossRefGoogle Scholar
[12]Hoehnke, H.-J.Konstructive methoden in der theorie der algebren. I. Monatsb. Deutsch. Akad. Wiss. Berlin 2 (1960), 138–146.Google Scholar
[13]Huppert, B.Endliche Gruppen I (Springer, Berlin, New York, 1967).CrossRefGoogle Scholar
[14]Johnson, K. W.On the group determinant. Math. Proc. Camb. Phil. Soc. 109 (1991), 299–311.CrossRefGoogle Scholar
[15]Johnson, K. W.The group determinant and k-characters, a survey and a conjecture, in Group Theory, Proceedings of the second biennial Ohio State-Denison Conference (Sehgal–Solomon Eds.) (World Scientific Press, Singapore, 1993), 181–197.Google Scholar
[16]Johnson, K. W.The Dedekind-Frobenius group determinant, new life in an old method. Proceedings Groups Bath (1997). London Math. Soc. Lecture Notes 261 (1999), 429–435.Google Scholar
[17]Lam, T. Y.Representations of finite groups: a hundred years, part I. Not. Amer. Math. Soc. 45, no. 3 (1998), 361–372.Google Scholar
[18]Taylor, R. L.Galois representations associated to Siegel modular forms of low weight. Duke Math. J. 63 (1991), 281–332.CrossRefGoogle Scholar
[19]Vazirani, M. J.Extending Frobenius' higher characters. Sci. Math. Japan. 58 (2003), 169–182.Google Scholar
[20]Wiles, A.On ordinary λ-adic representations associated to modular forms. Invent. Math. 94 (1988), 529–573.CrossRefGoogle Scholar