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Representations of wreath product

3

1

$begingroup$

I would like to know if there is a systematic way to decompose direct products of irreps of a wreath product of the orthogonal group. I'm talking about the group $G_m,n=O(m)wr S_n=O(m)^nrtimes S_n$. For example, I know that $G_m,n$ has an irrep of dimension $mn$ furnished by a vector $phi_i$, and an irrep of dimension $n-1$ furnished by a traceless-symmetric matrix $X_ij$, where $i,j=1,ldots,mn$. Is there a systematic way to construct the decomposition of the direct product of these two irreps, i.e. to compute $phi_iotimesphi_j$, $phi_iotimes X_jk$ and $X_ijotimes X_kl$?

For a wreath product of the form $G_n=Gammawr S_n=Gamma^nrtimes S_n$ with $Gamma$ a finite group the answer to the question above appears to be affirmative, but it relies on the fact that $Gamma$, being a finite group, has a finite number of irreps.

group-theory representation-theory

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asked Mar 26 at 16:53

AndySAndyS

1163

$endgroup$

add a comment | 

3

1

$begingroup$

I would like to know if there is a systematic way to decompose direct products of irreps of a wreath product of the orthogonal group. I'm talking about the group $G_m,n=O(m)wr S_n=O(m)^nrtimes S_n$. For example, I know that $G_m,n$ has an irrep of dimension $mn$ furnished by a vector $phi_i$, and an irrep of dimension $n-1$ furnished by a traceless-symmetric matrix $X_ij$, where $i,j=1,ldots,mn$. Is there a systematic way to construct the decomposition of the direct product of these two irreps, i.e. to compute $phi_iotimesphi_j$, $phi_iotimes X_jk$ and $X_ijotimes X_kl$?

For a wreath product of the form $G_n=Gammawr S_n=Gamma^nrtimes S_n$ with $Gamma$ a finite group the answer to the question above appears to be affirmative, but it relies on the fact that $Gamma$, being a finite group, has a finite number of irreps.

group-theory representation-theory

share|cite|improve this question

asked Mar 26 at 16:53

AndySAndyS

1163

$endgroup$

add a comment | 

$begingroup$

I would like to know if there is a systematic way to decompose direct products of irreps of a wreath product of the orthogonal group. I'm talking about the group $G_m,n=O(m)wr S_n=O(m)^nrtimes S_n$. For example, I know that $G_m,n$ has an irrep of dimension $mn$ furnished by a vector $phi_i$, and an irrep of dimension $n-1$ furnished by a traceless-symmetric matrix $X_ij$, where $i,j=1,ldots,mn$. Is there a systematic way to construct the decomposition of the direct product of these two irreps, i.e. to compute $phi_iotimesphi_j$, $phi_iotimes X_jk$ and $X_ijotimes X_kl$?

For a wreath product of the form $G_n=Gammawr S_n=Gamma^nrtimes S_n$ with $Gamma$ a finite group the answer to the question above appears to be affirmative, but it relies on the fact that $Gamma$, being a finite group, has a finite number of irreps.

group-theory representation-theory

share|cite|improve this question

asked Mar 26 at 16:53

AndySAndyS

1163

$endgroup$

share|cite|improve this question

asked Mar 26 at 16:53

AndySAndyS

1163

share|cite|improve this question

asked Mar 26 at 16:53

AndySAndyS

1163

asked Mar 26 at 16:53

AndySAndyS

1163

asked Mar 26 at 16:53

AndySAndyS

1163