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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?
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    $\begingroup$ The character of the identity element is the trace of the identity matrix, i.e. the dimension (the "degree") of the representation. $\endgroup$ Apr 15, 2023 at 21:52
  • $\begingroup$ The trace of the identity for any finite dimensional vector space is the sum of $1$s. $\endgroup$ Apr 15, 2023 at 21:52
  • $\begingroup$ Ah, I see. Thank you. $\endgroup$
    – Tensor
    Apr 15, 2023 at 22:06
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    $\begingroup$ @AnneBauval Can you make that into an answer? Although it is trivial, we do not want answers to be lost in comments. $\endgroup$
    – Pedro
    Apr 17, 2023 at 8:48
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    $\begingroup$ @AnneBauval No problem. $\endgroup$
    – Pedro
    Apr 17, 2023 at 9:06

1 Answer 1

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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.

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    $\begingroup$ Upvoting this. We have likelyly handled this earlier, but it is not as frequent a duplicate as most others. And I don't have the time to search either :-) $\endgroup$ Apr 17, 2023 at 9:14

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