# Character groups of $S_n$

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?

## 1 Answer

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.

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