I'm a chemist who is interested in mathematics. One of the subject that I'm interested in is group theory and representation theory, because of its application in chemistry.
I was looking at character tables and realize that the characters of identity element is always a positive integer (i.e. in $Sym(3)$ the character for identity element is 1 and 2, in $Sym(4)$ the character for identity element is 1, 2, and 3, etc.). Is there any reason for why this is true?
2026-03-27 00:09:38.1774570178
Why does character of identity element always positive integer?
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The character of the identity element is the trace of the identity matrix, i.e. the dimension (the "degree") of the representation.