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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.25 no.3 Antofagasta Dec. 2006

http://dx.doi.org/10.4067/S0716-09172006000300007 

 

Proyecciones
Vol. 25, No 3, pp. 307-329, December 2006.
Universidad Católica del Norte
Antofagasta - Chile

 

THE COMPLEX LINEAR REPRESENTATIONS OF GL(2, k), k A FINITE FIELD

 

LUISA ABURTO*, ROBERTO JOHNSON †‡, JOSÉ PANTOJA §¶

P. UNIVERSIDAD CATÓLICA DE VALPARAÍSO, CHILE


Abstract

Let k be a finite field of odd characteristic, and let G be the group of all invertible 2 ¡¿ 2 matrices over k. We construct the irreducible complex linear representations of the group G.The constructions lean on the method of induction from subgroups and on the theory of characters. To accomplish this goal, the basic facts from the theory of representations and characters of finite groups are presented. Furthermore, we describe the structure of G that we need, and the theory of representations of some subgroups of G that we use. As a final result, we obtain the theory of the irreducible representations of G,by describing either the irreducible representations of , or the irreducible characters of the group G.


 

REFERENCES

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[Cu-Re] C. Curtis and I. Reiner, Representation Theory of Finite Groupsand Associative Algebras, Willey, N. York, (1962).

[G] S. Gelfand, Representations of the General Linear Group over a Finite Field, Mat. Sb. 83 1970, pp. 15-41, (1970).

[PS] I. Piatetski-Shapiro, Complex Representations of GL(2, k) for FiniteFields k, Contemporary Mathematics, Volume 16, AMS,(1983).

[S] K. Spindler, Abstract Algebra with Applications, Dekker, (1994).

[SA1] J. Soto-Andrade, Métodos Geométricos en Teoría de Representacionesde Grupos Finitos, IX Escuela Latinoamericana de Matemáticas (ELAM), Santiago-Chile, (1988).

[SA2] J. Soto-Andrade, Représentations de Certain Groupes Symplectiques Finis, Bull. Soc. Math. France, pp. 5-334, (1978).

[Se] J.P. Serre, Linear Representations of Finite Groups, Springer-Verlag, (1977).

[T] A. Terras, Fourier Analysis on Finite Groups and Applications, L. M. S. 43, Cambridge University Press, (2001).

 

Luisa Aburto-Hageman
Instituto de Matemáticas
Pontificia Universidad Católica de Valparaíso
Casilla 4059
Valparaíso
CHILE
email: laburto@ucv.cl

Roberto Johnson
Instituto de Matemáticas
Pontificia Universidad Católica de Valparaíso
Casilla 4059
Valparaíso
CHILE
email: rjohnson@ucv.cl

and

José Pantoja
Instituto de Matemáticas
Pontificia Universidad Católica de Valparaíso
Casilla 4059
Valparaíso
CHILE
email: jpantoja@ucv.cl

 

Received : October 2006. Accepted : November 2006

Partially supported by Pontificia Universidad Católica de Valparaíso DI 124.709/2006

Partially supported by Pontificia Universidad Católica de Valparaíso DI 124.708/2006
Partially supported by Universidad Católica del Norte, as a visiting professor
§Partially supported by FONDECYT 106051 and P. Universidad Católica de Valparaiso
Partially supported by Universidad Católica del Norte, as a visiting professor

 

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