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Math Help - Show that the matrix representation of the dihedral group D4 by M is irreducible.

  1. #1
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    Show that the matrix representation of the dihedral group D4 by M is irreducible.

    Show that the matrix representation of the dihedral group D4 by M is irreducible.

    You are given that all of the elements of a matrix group M can be generated
    from the following two elements,

    A=
    |0 -1|
    |1 0|

    B=
    |1 0|
    |0 -1|

    in the sense that all other elements can be written A^n B^m for integer m, n >or= 0.
    Find the remaining elements in M.
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  2. #2
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    Re: Show that the matrix representation of the dihedral group D4 by M is irreducible.

    Quote Originally Posted by blueyellow View Post
    Show that the matrix representation of the dihedral group D4 by M is irreducible.

    You are given that all of the elements of a matrix group M can be generated
    from the following two elements,

    A=
    |0 -1|
    |1 0|

    B=
    |1 0|
    |0 -1|

    in the sense that all other elements can be written A^n B^m for integer m, n >or= 0.
    Find the remaining elements in M.
    the representation is irreducible because A and B have no common eigenvectors. see the theorem in my blog.
    you should do the second part of the problem yourself. note that 0 \leq m \leq 1 and 0 \leq n \leq 3.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Re: Show that the matrix representation of the dihedral group D4 by M is irreducible.

    Quote Originally Posted by blueyellow View Post
    Show that the matrix representation of the dihedral group D4 by M is irreducible.

    You are given that all of the elements of a matrix group M can be generated
    from the following two elements,

    A=
    |0 -1|
    |1 0|

    B=
    |1 0|
    |0 -1|

    in the sense that all other elements can be written A^n B^m for integer m, n >or= 0.
    Find the remaining elements in M.
    Alternatively, you could do the second part first (which is trivial) and then (assuming you are talking about \mathbb{C}-representations) appeal to the common theorem that a representation is irreducible if and only if the inner product of the character with itself is one, i.e. if \displaystyle \frac{1}{|G|}\sum_{g\in G}|\chi_M(g)|^2=1.
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