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Math Help - Cycle index for rotation

  1. #1
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    Cycle index for rotation

    Consider a square where there are points on the four corners and the four midpoints. Find \hat{C}(r,b) using the cycle index for the group D4.

    The identity has 8 cycles of length 1; the three rotations have 2 cycles of length 4; the reflections have 2 cycles of length 1 and 3 cycles of length 2.
    Hence, the cycle index is:
    (z_{1}^{8}+3z_{4}^{2}+4z_{1}^{2}z_{2}^{3})/8
    So \hat{C}(r,b)=((r+b)^8+3(r^4+b^4)^2+4(r+b)^2(r^2+b^  2)^3)/8
    But when I expanded this using Wolfram Alpha, I got fractions for some of the coefficients.
    I can't seem to find what I did wrong, so I need help finding out.
    Thanks in advance.
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  2. #2
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    Athens, OH, USA
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    Re: Cycle index for rotation

    Hi,
    What are "C hat", b and r? I'm totally ignorant about the usage of the cycle index, but I do know about the dihedral group. Rotation of the square through 180 degrees does not yield a product of two 4 cycles, but a permutation which is the product of four 2 cycles. See the attachment for the computation of the cycle index.

    Cycle index for rotation-mhfcycleindex.png
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  3. #3
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    Re: Cycle index for rotation

    The C hat computes the number of possible colorings of corners with different distributions of red and blue.
    But your answer showed that I made a mistake in my calculation; thank you.
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