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I am trying to understand if we can define the concept of character group for the irreps of the symmetric group $S_n$. Considering a few examples like the character tables of $S_3$, $S_5$, and $S_6$ it looks to me that the collection of the elements of the character table of a symmetric group misses at least the closure property if the group operation is multiplication.

Can I assume the followings:

  • it is not possible to construct a group from the elements of the character table of a symmetric group?
  • it is not possible to construct a group from the elements of the character table of a symmetric group with the operation as multiplication modulo some integer?
  • it is not possible to construct a group from the characters of a given representation of a symmetric group?
  • it is not possible to construct a group from the characters of a given representation of a symmetric group with the operation as multiplication modulo some integer?
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Helas there is no way to construct a group structure on characters/representations. If you know that the image of the identity by a character is the dimension of the representation, e.g. $3$, how would you define a representation on a vector space of dimension $1/3$ ? The good news is that the set of characters can be turned into a ring. See ring of characters for more details.

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  • $\begingroup$ why do I need to define a representation on a vector space of fractional dimension? $\endgroup$ – Omar Shehab Dec 21 '16 at 7:45

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