Mixing
Irreducible Representations of $S4$
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so ive been trying to get all the irreducible representations of $S4$ and their dimensions, I know that there are 5 because they are 5 conjugacy classes in $S4$ and that 2 have dimension 1 because $S4/[S4,S4] cong mathbb{Z}_2$, but i need to find the dimension of the other tree, where i tried using the formula $|S4|=1+1+n_3^2+n_4^2+n_5^2$. My question is is there any other way to find out the dimensions of these representations without having to find the right 3 numbers that squared give 22? Is there another way to do it ? Thanks.
abstract-algebra representation-theory
asked Jan 26 at 14:37
Pedro SantosPedro Santos
1539
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|
show 5 more comments
$begingroup$
so ive been trying to get all the irreducible representations of $S4$ and their dimensions, I know that there are 5 because they are 5 conjugacy classes in $S4$ and that 2 have dimension 1 because $S4/[S4,S4] cong mathbb{Z}_2$, but i need to find the dimension of the other tree, where i tried using the formula $|S4|=1+1+n_3^2+n_4^2+n_5^2$. My question is is there any other way to find out the dimensions of these representations without having to find the right 3 numbers that squared give 22? Is there another way to do it ? Thanks.
abstract-algebra representation-theory
asked Jan 26 at 14:37
Pedro SantosPedro Santos
1539
$endgroup$
$begingroup$
Hint: $S_4$ acts on a $4$-dimensional vector space by permuting basis vectors. Can you check if this is irreducible, or if not how it decomposes? Be careful to consider special cases on the characteristic of the field.
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– Adam Higgins
Jan 26 at 14:41
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I'd consider the Specht modules.
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– Wuestenfux
Jan 26 at 14:44
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@AdamHiggins Seeing as he says there are $5$, he is presumably working over the complex numbers. But your suggestion is still a great place to start since it finds one of the numbers.
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– Tobias Kildetoft
Jan 26 at 14:52
1
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@AdamHiggins Sure, and in those cases you don't need to worry about considerations of characteristic (but the way it is phrased makes me expect this is just for complex numbers).
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– Tobias Kildetoft
Jan 26 at 14:58
1
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@Luca the equation written is not the class equation, so it is not related to the center.
$endgroup$
– Tobias Kildetoft
Jan 26 at 18:05
|
show 5 more comments
$begingroup$
so ive been trying to get all the irreducible representations of $S4$ and their dimensions, I know that there are 5 because they are 5 conjugacy classes in $S4$ and that 2 have dimension 1 because $S4/[S4,S4] cong mathbb{Z}_2$, but i need to find the dimension of the other tree, where i tried using the formula $|S4|=1+1+n_3^2+n_4^2+n_5^2$. My question is is there any other way to find out the dimensions of these representations without having to find the right 3 numbers that squared give 22? Is there another way to do it ? Thanks.
abstract-algebra representation-theory
asked Jan 26 at 14:37
Pedro SantosPedro Santos
1539
$endgroup$
so ive been trying to get all the irreducible representations of $S4$ and their dimensions, I know that there are 5 because they are 5 conjugacy classes in $S4$ and that 2 have dimension 1 because $S4/[S4,S4] cong mathbb{Z}_2$, but i need to find the dimension of the other tree, where i tried using the formula $|S4|=1+1+n_3^2+n_4^2+n_5^2$. My question is is there any other way to find out the dimensions of these representations without having to find the right 3 numbers that squared give 22? Is there another way to do it ? Thanks.
abstract-algebra representation-theory
abstract-algebra representation-theory
asked Jan 26 at 14:37
Pedro SantosPedro Santos
1539
asked Jan 26 at 14:37
Pedro SantosPedro Santos
1539
asked Jan 26 at 14:37
Pedro SantosPedro Santos
1539
asked Jan 26 at 14:37
Pedro SantosPedro Santos
1539
asked Jan 26 at 14:37
Pedro SantosPedro Santos
1539
1539
$begingroup$
Hint: $S_4$ acts on a $4$-dimensional vector space by permuting basis vectors. Can you check if this is irreducible, or if not how it decomposes? Be careful to consider special cases on the characteristic of the field.
$endgroup$
– Adam Higgins
Jan 26 at 14:41
$begingroup$
I'd consider the Specht modules.
$endgroup$
– Wuestenfux
Jan 26 at 14:44
$begingroup$
@AdamHiggins Seeing as he says there are $5$, he is presumably working over the complex numbers. But your suggestion is still a great place to start since it finds one of the numbers.
$endgroup$
– Tobias Kildetoft
Jan 26 at 14:52
1
$begingroup$
@AdamHiggins Sure, and in those cases you don't need to worry about considerations of characteristic (but the way it is phrased makes me expect this is just for complex numbers).
$endgroup$
– Tobias Kildetoft
Jan 26 at 14:58
1
$begingroup$
@Luca the equation written is not the class equation, so it is not related to the center.
$endgroup$
– Tobias Kildetoft
Jan 26 at 18:05
|
show 5 more comments
$begingroup$
Hint: $S_4$ acts on a $4$-dimensional vector space by permuting basis vectors. Can you check if this is irreducible, or if not how it decomposes? Be careful to consider special cases on the characteristic of the field.
$endgroup$
– Adam Higgins
Jan 26 at 14:41
$begingroup$
I'd consider the Specht modules.
$endgroup$
– Wuestenfux
Jan 26 at 14:44
$begingroup$
@AdamHiggins Seeing as he says there are $5$, he is presumably working over the complex numbers. But your suggestion is still a great place to start since it finds one of the numbers.
$endgroup$
– Tobias Kildetoft
Jan 26 at 14:52
1
$begingroup$
@AdamHiggins Sure, and in those cases you don't need to worry about considerations of characteristic (but the way it is phrased makes me expect this is just for complex numbers).
$endgroup$
– Tobias Kildetoft
Jan 26 at 14:58
1
$begingroup$
@Luca the equation written is not the class equation, so it is not related to the center.
$endgroup$
– Tobias Kildetoft
Jan 26 at 18:05
$begingroup$
Hint: $S_4$ acts on a $4$-dimensional vector space by permuting basis vectors. Can you check if this is irreducible, or if not how it decomposes? Be careful to consider special cases on the characteristic of the field.
$endgroup$
– Adam Higgins
Jan 26 at 14:41
$begingroup$
Hint: $S_4$ acts on a $4$-dimensional vector space by permuting basis vectors. Can you check if this is irreducible, or if not how it decomposes? Be careful to consider special cases on the characteristic of the field.
$endgroup$
– Adam Higgins
Jan 26 at 14:41
$begingroup$
I'd consider the Specht modules.
$endgroup$
– Wuestenfux
Jan 26 at 14:44
$begingroup$
I'd consider the Specht modules.
$endgroup$
– Wuestenfux
Jan 26 at 14:44
$begingroup$
@AdamHiggins Seeing as he says there are $5$, he is presumably working over the complex numbers. But your suggestion is still a great place to start since it finds one of the numbers.
$endgroup$
– Tobias Kildetoft
Jan 26 at 14:52
$begingroup$
@AdamHiggins Seeing as he says there are $5$, he is presumably working over the complex numbers. But your suggestion is still a great place to start since it finds one of the numbers.
$endgroup$
– Tobias Kildetoft
Jan 26 at 14:52
1
1
$begingroup$
@AdamHiggins Sure, and in those cases you don't need to worry about considerations of characteristic (but the way it is phrased makes me expect this is just for complex numbers).
$endgroup$
– Tobias Kildetoft
Jan 26 at 14:58
$begingroup$
@AdamHiggins Sure, and in those cases you don't need to worry about considerations of characteristic (but the way it is phrased makes me expect this is just for complex numbers).
$endgroup$
– Tobias Kildetoft
Jan 26 at 14:58
1
1
$begingroup$
@Luca the equation written is not the class equation, so it is not related to the center.
$endgroup$
– Tobias Kildetoft
Jan 26 at 18:05
$begingroup$
@Luca the equation written is not the class equation, so it is not related to the center.
$endgroup$
– Tobias Kildetoft
Jan 26 at 18:05
|
show 5 more comments
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$begingroup$
Hint: $S_4$ acts on a $4$-dimensional vector space by permuting basis vectors. Can you check if this is irreducible, or if not how it decomposes? Be careful to consider special cases on the characteristic of the field.
$endgroup$
– Adam Higgins
Jan 26 at 14:41
$begingroup$
I'd consider the Specht modules.
$endgroup$
– Wuestenfux
Jan 26 at 14:44
$begingroup$
@AdamHiggins Seeing as he says there are $5$, he is presumably working over the complex numbers. But your suggestion is still a great place to start since it finds one of the numbers.
$endgroup$
– Tobias Kildetoft
Jan 26 at 14:52
1
$begingroup$
@AdamHiggins Sure, and in those cases you don't need to worry about considerations of characteristic (but the way it is phrased makes me expect this is just for complex numbers).
$endgroup$
– Tobias Kildetoft
Jan 26 at 14:58
1
$begingroup$
@Luca the equation written is not the class equation, so it is not related to the center.
$endgroup$
– Tobias Kildetoft
Jan 26 at 18:05