Complex Representations of A4












2












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Im asked to obtain the number of irreducible complex representations of the group $A_4$ and their repesctive dimensions. I know that the number of irreducible representations is going to be the number of conjugacy classes , so there are 4, but how do i obtain they re dimension? I also know that $|G|= sum_{i=1}^{n} n_i^2$ where the $n_i$ are the dimension but i dont see how that helps me. Thanks.






abstract-algebra representation-theory







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  • $begingroup$
    I'd start with the degree one representations.
    $endgroup$
    – Lord Shark the Unknown
    Jan 22 at 19:54










  • $begingroup$
    I guess i would use the fact that there is a bijection between the representations of degree 1 of $A_4$ and $A_4/[A_4,A_4]$, now just got determine what that is.
    $endgroup$
    – Pedro Santos
    Jan 22 at 19:57
















2












$begingroup$


Im asked to obtain the number of irreducible complex representations of the group $A_4$ and their repesctive dimensions. I know that the number of irreducible representations is going to be the number of conjugacy classes , so there are 4, but how do i obtain they re dimension? I also know that $|G|= sum_{i=1}^{n} n_i^2$ where the $n_i$ are the dimension but i dont see how that helps me. Thanks.






abstract-algebra representation-theory







share|cite|improve this question









$endgroup$












  • $begingroup$
    I'd start with the degree one representations.
    $endgroup$
    – Lord Shark the Unknown
    Jan 22 at 19:54










  • $begingroup$
    I guess i would use the fact that there is a bijection between the representations of degree 1 of $A_4$ and $A_4/[A_4,A_4]$, now just got determine what that is.
    $endgroup$
    – Pedro Santos
    Jan 22 at 19:57














2












2








2





$begingroup$


Im asked to obtain the number of irreducible complex representations of the group $A_4$ and their repesctive dimensions. I know that the number of irreducible representations is going to be the number of conjugacy classes , so there are 4, but how do i obtain they re dimension? I also know that $|G|= sum_{i=1}^{n} n_i^2$ where the $n_i$ are the dimension but i dont see how that helps me. Thanks.






abstract-algebra representation-theory







share|cite|improve this question









$endgroup$




Im asked to obtain the number of irreducible complex representations of the group $A_4$ and their repesctive dimensions. I know that the number of irreducible representations is going to be the number of conjugacy classes , so there are 4, but how do i obtain they re dimension? I also know that $|G|= sum_{i=1}^{n} n_i^2$ where the $n_i$ are the dimension but i dont see how that helps me. Thanks.






abstract-algebra representation-theory




abstract-algebra representation-theory






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 22 at 19:52









Pedro SantosPedro Santos

1539




1539












  • $begingroup$
    I'd start with the degree one representations.
    $endgroup$
    – Lord Shark the Unknown
    Jan 22 at 19:54










  • $begingroup$
    I guess i would use the fact that there is a bijection between the representations of degree 1 of $A_4$ and $A_4/[A_4,A_4]$, now just got determine what that is.
    $endgroup$
    – Pedro Santos
    Jan 22 at 19:57


















  • $begingroup$
    I'd start with the degree one representations.
    $endgroup$
    – Lord Shark the Unknown
    Jan 22 at 19:54










  • $begingroup$
    I guess i would use the fact that there is a bijection between the representations of degree 1 of $A_4$ and $A_4/[A_4,A_4]$, now just got determine what that is.
    $endgroup$
    – Pedro Santos
    Jan 22 at 19:57
















$begingroup$
I'd start with the degree one representations.
$endgroup$
– Lord Shark the Unknown
Jan 22 at 19:54




$begingroup$
I'd start with the degree one representations.
$endgroup$
– Lord Shark the Unknown
Jan 22 at 19:54












$begingroup$
I guess i would use the fact that there is a bijection between the representations of degree 1 of $A_4$ and $A_4/[A_4,A_4]$, now just got determine what that is.
$endgroup$
– Pedro Santos
Jan 22 at 19:57




$begingroup$
I guess i would use the fact that there is a bijection between the representations of degree 1 of $A_4$ and $A_4/[A_4,A_4]$, now just got determine what that is.
$endgroup$
– Pedro Santos
Jan 22 at 19:57










1 Answer
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$begingroup$

Hint As OP suggests in the comments, one can start by identifying $$A_4 / [A_4 , A_4] cong A_4 / (Bbb Z_2 times Bbb Z_2) cong Bbb Z_3 ,$$ which gives immediately that there are $3$ representations of dimension $1$, and so by the sum-of-squares formula $1$ representation of dimension $3$.



Alternatively, one can get away with using only the count of the conjugacy classes (which you've already found to be $4$) and the sum-of-squares formula, $$12 = |A_4| = sum_{i = 1}^4 n_i^2 ,$$
where $n_i$ is the dimension of the $i$th irreducible representation. Checking manually shows that the only way to write $12$ as a sum of four positive squares is $1^2 + 1^2 + 1^2 + 3^2$. Of course, the existence of the trivial representation means that we need only look for ways to write $11$ as a sum of three squares.






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    $begingroup$

    Hint As OP suggests in the comments, one can start by identifying $$A_4 / [A_4 , A_4] cong A_4 / (Bbb Z_2 times Bbb Z_2) cong Bbb Z_3 ,$$ which gives immediately that there are $3$ representations of dimension $1$, and so by the sum-of-squares formula $1$ representation of dimension $3$.



    Alternatively, one can get away with using only the count of the conjugacy classes (which you've already found to be $4$) and the sum-of-squares formula, $$12 = |A_4| = sum_{i = 1}^4 n_i^2 ,$$
    where $n_i$ is the dimension of the $i$th irreducible representation. Checking manually shows that the only way to write $12$ as a sum of four positive squares is $1^2 + 1^2 + 1^2 + 3^2$. Of course, the existence of the trivial representation means that we need only look for ways to write $11$ as a sum of three squares.






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      Hint As OP suggests in the comments, one can start by identifying $$A_4 / [A_4 , A_4] cong A_4 / (Bbb Z_2 times Bbb Z_2) cong Bbb Z_3 ,$$ which gives immediately that there are $3$ representations of dimension $1$, and so by the sum-of-squares formula $1$ representation of dimension $3$.



      Alternatively, one can get away with using only the count of the conjugacy classes (which you've already found to be $4$) and the sum-of-squares formula, $$12 = |A_4| = sum_{i = 1}^4 n_i^2 ,$$
      where $n_i$ is the dimension of the $i$th irreducible representation. Checking manually shows that the only way to write $12$ as a sum of four positive squares is $1^2 + 1^2 + 1^2 + 3^2$. Of course, the existence of the trivial representation means that we need only look for ways to write $11$ as a sum of three squares.






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        Hint As OP suggests in the comments, one can start by identifying $$A_4 / [A_4 , A_4] cong A_4 / (Bbb Z_2 times Bbb Z_2) cong Bbb Z_3 ,$$ which gives immediately that there are $3$ representations of dimension $1$, and so by the sum-of-squares formula $1$ representation of dimension $3$.



        Alternatively, one can get away with using only the count of the conjugacy classes (which you've already found to be $4$) and the sum-of-squares formula, $$12 = |A_4| = sum_{i = 1}^4 n_i^2 ,$$
        where $n_i$ is the dimension of the $i$th irreducible representation. Checking manually shows that the only way to write $12$ as a sum of four positive squares is $1^2 + 1^2 + 1^2 + 3^2$. Of course, the existence of the trivial representation means that we need only look for ways to write $11$ as a sum of three squares.






        share|cite|improve this answer









        $endgroup$



        Hint As OP suggests in the comments, one can start by identifying $$A_4 / [A_4 , A_4] cong A_4 / (Bbb Z_2 times Bbb Z_2) cong Bbb Z_3 ,$$ which gives immediately that there are $3$ representations of dimension $1$, and so by the sum-of-squares formula $1$ representation of dimension $3$.



        Alternatively, one can get away with using only the count of the conjugacy classes (which you've already found to be $4$) and the sum-of-squares formula, $$12 = |A_4| = sum_{i = 1}^4 n_i^2 ,$$
        where $n_i$ is the dimension of the $i$th irreducible representation. Checking manually shows that the only way to write $12$ as a sum of four positive squares is $1^2 + 1^2 + 1^2 + 3^2$. Of course, the existence of the trivial representation means that we need only look for ways to write $11$ as a sum of three squares.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 22 at 21:37









        TravisTravis

        63k767150




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