Results 1 to 2 of 2

Math Help - Symmetry "decomposition" of functions

  1. #1
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,901
    Thanks
    329
    Awards
    1

    Symmetry "decomposition" of functions

    First, sorry for the incorrect terminology, I'm not sure what else to call this.

    I know it is possible to use the group C_2 to show that a function f(x) breaks down into the sum of an even and an odd function. Since I can't find my notes or reference on the subject my questions are
    1) How do you do this using C_2
    and
    2) How do you generalize this process? As an example, how can you do this with, say, D_3?

    Thanks!
    -Dan
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,901
    Thanks
    329
    Awards
    1
    Thank you to any who have looked at this problem.

    I finally found the text reference. I'll show how to do the C_2 problem. The D_3 problem can be done similarly, but I have yet to manage to construct any useful basis functions for it.

    The relevant equation to find functions with symmetries under a particular irreducible group representation is
    \sum_R \chi ^{(\mu)*}(R)O_R \psi^{( \mu )} = \frac{g}{n_{\mu}} \psi ^{(\mu)}
    where the sum is over the group elements, \chi(R) is the character of the group element R, O_R is the operator (corresponding to the group element R) that acts in function space, \psi is a function, g is the order of the group, and n is the dimension of the representation under study. The superscript \mu indicates that we are working in the \muth representation.

    So applying this to the A representation of C_2 we get
    1 \cdot O_E f(x) + 1 \cdot O_{C_2} f(x) = \frac{2}{1}f(x)

    f(x) + f(-x) = 2f(x)

    f(x) = f(-x)

    So the A representation leads to basis functions that are even.

    Applying this to the B representation of C_2 we get
    1 \cdot O_E f(x) + (-1) \cdot O_{C_2} f(x) = \frac{2}{1}f(x)

    f(x) - f(-x) = 2f(x)

    f(x) = -f(-x)

    So the B representation leads to basis functions that are odd.

    Thus any function can be represented by the sum of the basis functions over all representations. In this case that means
    f(x) = f_A(x) + f_B(x)
    ie. any function may be written in terms of the sum of an even function and an odd function.

    This process can be applied to any group, though the interpretation of the basis functions can be a bit hairy for certain group symmetries.

    -Dan
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: October 17th 2011, 02:50 PM
  2. Replies: 1
    Last Post: September 16th 2011, 01:08 AM
  3. Replies: 2
    Last Post: June 4th 2011, 12:11 PM
  4. Replies: 2
    Last Post: April 24th 2011, 07:01 AM
  5. Replies: 1
    Last Post: October 25th 2010, 04:45 AM

Search Tags


/mathhelpforum @mathhelpforum