Irreducible representations of symmetric group












2












$begingroup$


Thanks to characters of representation we know that exists a bijection between irreducible representations of a finite group G and its conjugate classes.



That bijection is proved showing that the cardinality of the two sets is the same. (J-P Serre, Linear Representations of finite groups, Th. 7 pag 19)



Is there a canonical bijection? I think the answer it's no for a generic group, but it seems to me that for the symmetric group things could be different.






abstract-algebra representation-theory young-tableaux







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  • 3




    $begingroup$
    The keyword here is 'Young Tableaux'.
    $endgroup$
    – hellHound
    Dec 30 '18 at 17:17
















2












$begingroup$


Thanks to characters of representation we know that exists a bijection between irreducible representations of a finite group G and its conjugate classes.



That bijection is proved showing that the cardinality of the two sets is the same. (J-P Serre, Linear Representations of finite groups, Th. 7 pag 19)



Is there a canonical bijection? I think the answer it's no for a generic group, but it seems to me that for the symmetric group things could be different.






abstract-algebra representation-theory young-tableaux







share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    The keyword here is 'Young Tableaux'.
    $endgroup$
    – hellHound
    Dec 30 '18 at 17:17














2












2








2





$begingroup$


Thanks to characters of representation we know that exists a bijection between irreducible representations of a finite group G and its conjugate classes.



That bijection is proved showing that the cardinality of the two sets is the same. (J-P Serre, Linear Representations of finite groups, Th. 7 pag 19)



Is there a canonical bijection? I think the answer it's no for a generic group, but it seems to me that for the symmetric group things could be different.






abstract-algebra representation-theory young-tableaux







share|cite|improve this question











$endgroup$




Thanks to characters of representation we know that exists a bijection between irreducible representations of a finite group G and its conjugate classes.



That bijection is proved showing that the cardinality of the two sets is the same. (J-P Serre, Linear Representations of finite groups, Th. 7 pag 19)



Is there a canonical bijection? I think the answer it's no for a generic group, but it seems to me that for the symmetric group things could be different.






abstract-algebra representation-theory young-tableaux




abstract-algebra representation-theory young-tableaux






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 30 '18 at 17:29







ecrin

















asked Dec 30 '18 at 17:03









ecrinecrin

3477




3477








  • 3




    $begingroup$
    The keyword here is 'Young Tableaux'.
    $endgroup$
    – hellHound
    Dec 30 '18 at 17:17














  • 3




    $begingroup$
    The keyword here is 'Young Tableaux'.
    $endgroup$
    – hellHound
    Dec 30 '18 at 17:17








3




3




$begingroup$
The keyword here is 'Young Tableaux'.
$endgroup$
– hellHound
Dec 30 '18 at 17:17




$begingroup$
The keyword here is 'Young Tableaux'.
$endgroup$
– hellHound
Dec 30 '18 at 17:17










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$begingroup$

Yes, there's a canonical bijection. Namely, there's a canonical bijection between conjugacy classes in the symmetric group $S_n$ and partitions of $n$, and there's a canonical bijection between partitions of $n$ and irreducible representations. There are many references, but see for example Young Tableaux by Fulton.



The construction in Fulton is not the only possible construction. However, it is commented in the book that all of the various methods of constructing the representations known at this point end up with the same bijection with partitions. Arguably you could call it a coincidence, but seeing the construction in terms of Young tableaux makes it hard to think of it as a coincidence.






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    $begingroup$

    Yes, there's a canonical bijection. Namely, there's a canonical bijection between conjugacy classes in the symmetric group $S_n$ and partitions of $n$, and there's a canonical bijection between partitions of $n$ and irreducible representations. There are many references, but see for example Young Tableaux by Fulton.



    The construction in Fulton is not the only possible construction. However, it is commented in the book that all of the various methods of constructing the representations known at this point end up with the same bijection with partitions. Arguably you could call it a coincidence, but seeing the construction in terms of Young tableaux makes it hard to think of it as a coincidence.






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      Yes, there's a canonical bijection. Namely, there's a canonical bijection between conjugacy classes in the symmetric group $S_n$ and partitions of $n$, and there's a canonical bijection between partitions of $n$ and irreducible representations. There are many references, but see for example Young Tableaux by Fulton.



      The construction in Fulton is not the only possible construction. However, it is commented in the book that all of the various methods of constructing the representations known at this point end up with the same bijection with partitions. Arguably you could call it a coincidence, but seeing the construction in terms of Young tableaux makes it hard to think of it as a coincidence.






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        Yes, there's a canonical bijection. Namely, there's a canonical bijection between conjugacy classes in the symmetric group $S_n$ and partitions of $n$, and there's a canonical bijection between partitions of $n$ and irreducible representations. There are many references, but see for example Young Tableaux by Fulton.



        The construction in Fulton is not the only possible construction. However, it is commented in the book that all of the various methods of constructing the representations known at this point end up with the same bijection with partitions. Arguably you could call it a coincidence, but seeing the construction in terms of Young tableaux makes it hard to think of it as a coincidence.






        share|cite|improve this answer









        $endgroup$



        Yes, there's a canonical bijection. Namely, there's a canonical bijection between conjugacy classes in the symmetric group $S_n$ and partitions of $n$, and there's a canonical bijection between partitions of $n$ and irreducible representations. There are many references, but see for example Young Tableaux by Fulton.



        The construction in Fulton is not the only possible construction. However, it is commented in the book that all of the various methods of constructing the representations known at this point end up with the same bijection with partitions. Arguably you could call it a coincidence, but seeing the construction in terms of Young tableaux makes it hard to think of it as a coincidence.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 1 at 17:33









        Matt SamuelMatt Samuel

        38.6k63768




        38.6k63768






























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