Results 1 to 3 of 3

Math Help - Character/ Representation Theory

  1. #1
    Junior Member
    Joined
    Oct 2010
    Posts
    49

    Character/ Representation Theory

    Can anybody help me with these questions as I am completely lost and haven't done any character theory or rep theory for a long time


    (1) Let X be the matrix comprising the character table of a group.
    Show that X is non-singular & hence prove that the number of real-valued characters is equal to the number
    of self-inverse conjugacy classes.

    (2) Show that in a group G of odd order, no element other than the identity
    is conjugate to its inverse. Deduce that in a group of odd order, the trivial character is the only
    irreducible character that is real-valued.
    Last edited by Ackbeet; March 14th 2011 at 04:43 AM. Reason: Split threads so no thread has more than two problems.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    May 2010
    Posts
    95
    Quote Originally Posted by Turloughmack View Post
    Can anybody help me with these questions as I am completely lost and haven't done any character theory or rep theory for a long time


    (1) Let X be the matrix comprising the character table of a group.
    Show that X is non-singular & hence prove that the number of real-valued characters is equal to the number
    of self-inverse conjugacy classes.
    This is a sketch of the proof borrowed from the book "Representations and characters of groups" by Liebeck (p264).

    The irreducible character of G are linearly indepdent over \mathbb{C} as functions from G to \mathbb{C}, which follows that X is non-singular.

    Let \bar{X} be the complex conjugate of matrix X. For each irreducible character \chi^i, it is known that \bar{\chi^i} is also an irreducible character. Thus, we can construct a permutation matrix P induced from \chi^i \rightarrow \bar{\chi^i} such that PX=\bar{X}.

    Similarly, for each conjugacy class C_i(g_i), the entries in the column of X corresponding to C_i(g_i) are the complex conjugates of the entries in the column of X corresponding to \bar{C_i}(g_i^{-1}), where C_i denotes the conjugacy class of its representative g_i, and \bar{C_i} the conjugacy class of g_i^{-1}. Similarly, we can construct a permutation matrix Q induced from C_i \rightarrow \bar{C_i} such that XQ=\bar{X}. It follows that P=XQX^{-1}. Thus, we see that P and Q have the same trace. Recall that trace of a permutation matrix corresponds to the number of fixed points by the permutation. Therefore, the number of irreducible real-valued characters (corresponding to the fixed points of the first permutation) is the trace of P, and the number of self-inverse conjugacy classes (corresponding the the fixed points of the second permutation) is the trace of Q, which are equal.


    (2) Show that in a group G of odd order, no element other than the identity
    is conjugate to its inverse. Deduce that in a group of odd order, the trivial character is the only
    irreducible character that is real-valued.
    Note that C_i=\bar{C_i} iff each g_i is conjugate to g_i^{-1}, i.e., hg_ih^{-1}=g_i^{-1} for some h \in G. Then, h^2g_ih^{-2}=g_i, h^3g_ih^{-3}=g_i^{-1}, etc. Since G is of odd order, we see that g_i=g_i^{-1}(verify this). Thus, only the identity is conjugate to its inverse. Hence, the self-inverse conjugacy class is {1}. It follows that the number of irreducible real-valued characters is only one by (1).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Alternatively, one can appeal to the amazing theorem that for a finite group G if \sqrt{g}=\#\left\{h\in G:h^2=g\right\} then the number of self-conjugate representations of G is \displaystyle \frac{1}{|G|}\sum_{g\in G}\sqrt{g}. Although, the proof of this comes startling close to the ideas that the previous paster mentioned.
    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. Easy character theory proof
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 4th 2010, 01:56 PM
  4. Representation Theory
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 27th 2009, 08:01 AM
  5. Character Theory of Finite Groups- Applications
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: April 3rd 2009, 11:25 AM

Search Tags


/mathhelpforum @mathhelpforum