Results 1 to 3 of 3

Math Help - Character table: sum of elements in rows

  1. #1
    Newbie
    Joined
    Apr 2011
    Posts
    1

    Character table: sum of elements in rows

    Prove that the sum of elements in any row of the character table of a finite group G is a non-negative integer.

    This can be proven by considering the character of the permutation representation acquired by letting G act on itself by conjugation.

    I don't see the significance of the permutation character here. Any hints would be appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by bluepidgeon View Post
    Prove that the sum of elements in any row of the character table of a finite group G is a non-negative integer.

    This can be proven by considering the character of the permutation representation acquired by letting G act on itself by conjugation.

    I don't see the significance of the permutation character here. Any hints would be appreciated.
    I'm a little bit confused, about the hint. What you're basically saying is that if \chi\in\text{irr}(G) then \displaystyle \sum_{g\in G}\chi(g)\in\mathbb{N}\cup\{0\}, right? But, think about it if we let \chi^{\text{triv}} denote the trivial character g\mapsto 1\in\mathbb{C} then by the orthogonality relations for irreducible characters we have that \displaystyle \left\langle \chi,\chi^{\text{triv}}\right\rangle=\delta_{\chi,  \chi^{\text{triv}}. But, by definition \displaystyle \left\langle\chi,\chi^{\text{triv}}\right\rangle=\  frac{1}{|G|}\sum_{g\in G}\chi(g)\overline{\chi^{\text{triv}}(g)}=\frac{1}  {|G|}\sum_{g\in G}\chi(g) so that \displaystyle \sum_{g\in G}\chi(g)=\delta_{\chi,\chi^{\text{triv}}}|G|, which is what we want, right?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by Drexel28 View Post
    I'm a little bit confused, about the hint. What you're basically saying is that if \chi\in\text{irr}(G) then \displaystyle \sum_{g\in G}\chi(g)\in\mathbb{N}\cup\{0\}, right? But, think about it if we let \chi^{\text{triv}} denote the trivial character g\mapsto 1\in\mathbb{C} then by the orthogonality relations for irreducible characters we have that \displaystyle \left\langle \chi,\chi^{\text{triv}}\right\rangle=\delta_{\chi,  \chi^{\text{triv}}. But, by definition \displaystyle \left\langle\chi,\chi^{\text{triv}}\right\rangle=\  frac{1}{|G|}\sum_{g\in G}\chi(g)\overline{\chi^{\text{triv}}(g)}=\frac{1}  {|G|}\sum_{g\in G}\chi(g) so that \displaystyle \sum_{g\in G}\chi(g)=\delta_{\chi,\chi^{\text{triv}}}|G|, which is what we want, right?
    but \sum_{g \in G} \chi(g) is not the sum of elements in the row of the character table corresponding to \chi because your sum is over all elements of G and not over representative elements of each conjugacy class. this problem can be resolved by looking at the character of permutation representation, acting on G by conjugation, rather than the trivial character.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Character Theory
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 11th 2011, 05:34 PM
  2. Character Theory II
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 11th 2011, 05:30 PM
  3. Character/ Representation Theory
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: March 14th 2011, 04:00 PM
  4. Eight character passwords
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: May 16th 2010, 06:45 AM
  5. random character selection
    Posted in the Statistics Forum
    Replies: 2
    Last Post: May 10th 2009, 05:53 PM

Search Tags


/mathhelpforum @mathhelpforum