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2 - Representations of a Group

Published online by Cambridge University Press:  18 December 2009

Haruzo Hida
Affiliation:
University of California, Los Angeles
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Summary

Here we shall give a detailed exposition of a general theory of Group representations, pseudo-representations and their deformation. These results will be used later. The reader who knows the theory well can skip this chapter.

Group Representations

A representation of degree n of a group G is a group homomorphism of G into the group of invertible n × n matrices GLn(A) with coefficients in a commutative ring A. When the structure of the group is very complicated or when the group is very large, such as the absolute Galois group over ℚ, it is often easier to study representations rather than the group itself. In this section, we study the basic properties of group representations.

Coefficient rings

Any ring in this section is commutative with the identity 1 = 1A. If we refer to an algebra R, then R may not be commutative. A ring A is called local if there is only one maximal ideal mA in A. An A-module M is artinian (resp. noetherian) if the set of A-submodules of M satisfies the descending (resp. ascending) chain condition. If A itself as an A-module is artinian (resp. noetherian), we just call A artinian (resp. noetherian). For an artinian A-module M, if MM1 ⊃ ⊃ Mn = {0} is the maximal descending chain of, A-submodules, the number n is called the length of M and is written as ℓA(M). If A is an artinian ring, A is noetherian (Akizuki's theorem, [CRT] Theorem 1.3.2).

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Publisher: Cambridge University Press
Print publication year: 2000

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  • Representations of a Group
  • Haruzo Hida, University of California, Los Angeles
  • Book: Modular Forms and Galois Cohomology
  • Online publication: 18 December 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511526046.003
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  • Representations of a Group
  • Haruzo Hida, University of California, Los Angeles
  • Book: Modular Forms and Galois Cohomology
  • Online publication: 18 December 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511526046.003
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Representations of a Group
  • Haruzo Hida, University of California, Los Angeles
  • Book: Modular Forms and Galois Cohomology
  • Online publication: 18 December 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511526046.003
Available formats
×