Irreducible representations of symmetric group
$begingroup$
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
asked Dec 30 '18 at 17:03
ecrinecrin
3477
$endgroup$
add a comment |
$begingroup$
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
asked Dec 30 '18 at 17:03
ecrinecrin
3477
$endgroup$
3
$begingroup$
The keyword here is 'Young Tableaux'.
$endgroup$
– hellHound
Dec 30 '18 at 17:17
add a comment |
$begingroup$
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
asked Dec 30 '18 at 17:03
ecrinecrin
3477
$endgroup$
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
asked Dec 30 '18 at 17:03
ecrinecrin
3477
asked Dec 30 '18 at 17:03
ecrinecrin
3477
edited Dec 30 '18 at 17:29
ecrin
asked Dec 30 '18 at 17:03
ecrinecrin
3477
asked Dec 30 '18 at 17:03
ecrinecrin
3477
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
add a comment |
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
add a comment |
1 Answer
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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.
answered Jan 1 at 17:33
Matt SamuelMatt Samuel
38.6k63768
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add a comment |
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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.
answered Jan 1 at 17:33
Matt SamuelMatt Samuel
38.6k63768
$endgroup$
add a comment |
$begingroup$
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.
answered Jan 1 at 17:33
Matt SamuelMatt Samuel
38.6k63768
$endgroup$
add a comment |
$begingroup$
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.
answered Jan 1 at 17:33
Matt SamuelMatt Samuel
38.6k63768
$endgroup$
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.
answered Jan 1 at 17:33
Matt SamuelMatt Samuel
38.6k63768
answered Jan 1 at 17:33
Matt SamuelMatt Samuel
38.6k63768
answered Jan 1 at 17:33
Matt SamuelMatt Samuel
38.6k63768
answered Jan 1 at 17:33
Matt SamuelMatt Samuel
38.6k63768
38.6k63768
add a comment |
add a comment |
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$begingroup$
The keyword here is 'Young Tableaux'.
$endgroup$
– hellHound
Dec 30 '18 at 17:17