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Math Help - characters of S_5

  1. #1
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    characters of S_5

    Write the multiplication for the characters of S_5. That is, write every product of irreducible characters as a sum of irreducible characters (the table will be symmetric).
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  2. #2
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    Quote Originally Posted by akc2010 View Post
    Write the multiplication for the characters of S_5. That is, write every product of irreducible characters as a sum of irreducible characters (the table will be symmetric).
    Since S_5 has 7 conjugacy classes, it has 7 irreducible characters.

    -----------------------------
    Size 1 10 20 30 24 15 20
    Class 1, (12), (123), (1234), (12345), (12)(34), (12345)
    -----------------------------
    -----------------------------
    X1 1 1 1 1 1 1 1
    X2 1 -1 1 -1 1 1 -1
    X3 4 2 1 0 -1 0 -1
    X4 4 -2 1 0 -1 0 1
    X5 6 0 0 0 1 -2 0
    X6 5 1 -1 -1 0 1 1
    X7 5 -1 -1 1 0 1 -1

    X1 is the character for the trivial representation of S_5.
    X2 is the character for the alternating(sign) representation of S_5.
    X3 is the standard representation of S_5.
    X4 = X2 \cdotX3. It turns out to be an irreducible character of S_5.
    X5 = 1/2(X3(g)^2 - X3(g^2)),
    X_6,
    X_7 = X2 \cdotX5.

    These characters satisfy both "row orthogonality theorem" and "column orthogonal theorem" (link).

    Now you need to do some calculations and fill the table.
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