Direct product of faithful representations into direct sum

In summary, the conversation discusses the decomposition of the direct product of two irreducible representations of a finite group into a direct sum of irreducible representations. The possibility of generating every other irreducible representation by successive tensor products is also mentioned, with reference to a result by Molien. The conversation also addresses the existence of finite groups in which none of the irreducible representations are faithful, with examples such as Z/2Z x Z/2Z and groups with noncyclic center. The conversation concludes with a clarification about the necessary conditions for the existence of faithful irreducible representations.
  • #1
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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  • #2
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 [itex]V^{\otimes n}[/itex] 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 [itex]\chi[/itex] has [itex]\chi(g)=\chi(1)[/itex] for some [itex]g\neq 1[/itex] then necessarily [itex]g \in \ker \chi[/itex] and [itex]\chi[/itex] isn't faithful.

Thus there are lots of other examples, e.g. any noncyclic abelian group, and more generally any group with noncyclic center.
 
  • #3
Thank you so much! That was totally what I wanted to know.

PS: Yes, I should have written 'tensor product' instead of 'direct product'.
 
  • #4
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:).
 
  • #5
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.
 
  • #6
Yes, you're right - the center being cyclic is a necessary but by no means sufficient condition!
 

What is the direct product of faithful representations into direct sum?

The direct product of faithful representations into direct sum is a mathematical operation that combines two or more faithful representations (also known as group representations) into a larger representation. This operation is used in group theory to study the structure and properties of groups.

What is a faithful representation?

A faithful representation is a type of group representation where each element of the group is uniquely represented by a matrix or linear transformation. In other words, the representation preserves the group structure and no two elements are represented by the same matrix.

How is the direct product of faithful representations into direct sum different from regular direct product?

The main difference between the two is that the direct product of faithful representations into direct sum preserves the group structure, while regular direct product does not necessarily do so. In other words, the direct product of faithful representations into direct sum is a more specific operation that is used in the study of groups.

Why is the direct product of faithful representations into direct sum important?

This operation is important because it allows us to study the structure and properties of larger groups by breaking them down into smaller, more manageable groups. It also helps us understand the relationship between different group representations and how they interact with each other.

Can the direct product of faithful representations into direct sum be applied to any type of group?

Yes, this operation can be applied to any type of group, including finite and infinite groups. It is a general operation in group theory and is used to study the structure and properties of all types of groups.

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