Direct product of faithful representations into direct sum

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Discussion Overview

The discussion revolves around the properties of irreducible representations of finite groups, specifically focusing on the generation of irreducible representations through direct products and the existence of finite groups with no faithful irreducible representations. The scope includes theoretical aspects of representation theory.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants propose that starting from a single faithful irreducible representation, it may be possible to generate every other irreducible representation through successive direct products.
  • One participant suggests that the term "tensor products" might be more appropriate than "direct products," referencing a theorem by Molien that states every irreducible representation is contained within some tensor power of a faithful irreducible representation.
  • It is noted that there are finite groups, such as Z/2Z x Z/2Z, where none of the irreducible representations are faithful, with a method to identify this using character tables.
  • Another participant acknowledges the clarification regarding the terminology and expresses gratitude for the information provided.
  • There is a discussion about the implications of having a cyclic center in relation to the existence of faithful irreducible representations, with one participant questioning if the result holds when the center is the identity.
  • One participant confirms that having a cyclic center is necessary but not sufficient for the existence of faithful irreducible representations.

Areas of Agreement / Disagreement

Participants express differing views on the terminology used (direct vs. tensor products) and the implications of group properties on the existence of faithful irreducible representations. The discussion remains unresolved regarding the conditions under which faithful irreducible representations exist.

Contextual Notes

Participants highlight the importance of precise definitions and the potential for misunderstanding in the context of representation theory. There are unresolved questions about the implications of group structure on representation properties.

rkrsnan
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Direct product of two irreducible representations of a finite group can be decomposed into a direct sum of irreducible representations. So, starting from a single faithful irreducible representation, is it possible generate every other irreducible representation by successively taking direct products?

My second question is (if it makes sense), can we have a finite group in which none of the irreducible representations are faithful?

Thanks.
 
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rkrsnan said:
Direct product of two irreducible representations of a finite group can be decomposed into a direct sum of irreducible representations. So, starting from a single faithful irreducible representation, is it possible generate every other irreducible representation by successively taking direct products?
Do you mean to say tensor products here instead of direct products? If so, then the answer to your question is in some sense affirmative. A result due to Molien (sometimes called the Burnside-Molien theorem) says that every irreducible representation of a finite group is contained inside some tensor power V^{\otimes n} of a faithful irreducible representation V.

My second question is (if it makes sense), can we have a finite group in which none of the irreducible representations are faithful?
Yes. For example Z/2Z x Z/2Z doesn't have any. You can spot this by looking at the character table: if the the column corresponding to \chi has \chi(g)=\chi(1) for some g\neq 1 then necessarily g \in \ker \chi and \chi isn't faithful.

Thus there are lots of other examples, e.g. any noncyclic abelian group, and more generally any group with noncyclic center.
 
Thank you so much! That was totally what I wanted to know.

PS: Yes, I should have written 'tensor product' instead of 'direct product'.
 
No problem. By the way, the end of my first paragraph above should of course read "of a faithful representation V" and not "of a faithful irreducible representation V" (as there might not be such a V! :smile:).
 
Yep, understood.
About the cyclic center and having faithful irreducible reps, does this result work if the center is identity? I can find examples of groups in which center is identity; in some cases faithful irreps exist and in some others it doesn't.
 
Yes, you're right - the center being cyclic is a necessary but by no means sufficient condition!
 

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