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Math Help - irreducible representations of the quaternion group

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    irreducible representations of the quaternion group

    Find all the irreducible representations of the quaternion group and write its character table.
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    Quote Originally Posted by akc2010 View Post
    Find all the irreducible representations of the quaternion group and write its character table.
    Call the quaternion group Q. I will say irrep for an irreducible representation and rep for a representation.
    Character table:
    To get its character table first show that Q has 5 conjugacy classes.
    Then find all its 1 dimensional reps as given below..
    First prove that [Q,Q] = \{\pm 1\}. We know that all the one dimensional reps of Q can be obtained from the one dimensional reps of Q/[Q,Q] (which is isomorphic to Klein 4 group). I assume you know the one dimensional reps of Klein 4 group. (hint: Just try all feasible combination of 1 and -1 in the table).
    Then use sum of (character of identity)^2 = #Q = 8 to figure out the only non 1 dimensional irrep's dimension is 2.
    Now use orthogonality relations to fill the table. You will observe non 1 dimensional rep's character has a peculiar nature which will help you figure out the irrep
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