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An extension theorem for integral representations

Part of: Algebraic number theory: global fields Representation theory of groups

Published online by Cambridge University Press:  09 April 2009

Wolfgang Knapp  and
Peter Schmid
Wolfgang Knapp
Affiliation:
Universität Tübingen Mathematisches InstitutAuf der Morgenstelle 10 D-72076 Tübingen, Germany
Peter Schmid
Affiliation:
Universität Tübingen Mathematisches InstitutAuf der Morgenstelle 10 D-72076 Tübingen, Germany
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Abstract

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By a fundamental theorem of Brauer every irreducible character of a finite group G can be written in the field Q(εm) of mth roots of unity where m is the exponent of G. Is it always possible to find a matrix representation over its ring Z[εm] of integers? In the present paper it is shown that this holds true provided it is valid for the quasisimple groups. The reduction to such groups relies on a useful extension theorem for integral representations. Iwasawa theory on class groups of cyclotomic fields gives evidence that the answer is at least affirmative when the exponent is replaced by the order, and provides for a general qualitative result.

Keywords

Integral representationsClifford theoryclass groups.

MSC classification

Secondary: 20C10: Integral representations of finite groups 11R23: Iwasawa theory 11R33: Integral representations related to algebraic numbers; Galois module structure of rings of integers 20C34: Representations of sporadic groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 63 , Issue 1 , August 1997 , pp. 1 - 15
Copyright
Copyright © Australian Mathematical Society 1997

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