Mixing
Tensor methods for representations of SU(n)
$begingroup$
I'm familiar with the Young Tableaux method for finding irreducible representations of $mathrm{SU}(n)$ for $ninmathbb{N}$, and I also know of the tensor methods for finding irreducible representations of $mathrm{SU}(3)$.
Tensor methods for $mathrm{SU}(3)$ seem to rely heavily on the fact that irreducible representations of this group are identified by two natural numbers $(p,q)$, which define the covariance and contravariance of the tensor associated with that representation. But, if I'm not wrong, and, $n-1$ are needed to identify every irreducible representation of $mathrm{SU}(n)$. I have no proof of this but it makes sense if we look at the Young Tableaux method.
My questions, then, are:
Do tensor methods for finding irreducible representations of $mathrm{SU}(n)$ exist for $n > 3$?
If so, which are those?
If not, how is it that they exist for $n leq 3$, why do they work?
representation-theory lie-groups tensors
asked Jan 28 at 22:38
TeicDaunTeicDaun
31116
$endgroup$
add a comment |
$begingroup$
I'm familiar with the Young Tableaux method for finding irreducible representations of $mathrm{SU}(n)$ for $ninmathbb{N}$, and I also know of the tensor methods for finding irreducible representations of $mathrm{SU}(3)$.
Tensor methods for $mathrm{SU}(3)$ seem to rely heavily on the fact that irreducible representations of this group are identified by two natural numbers $(p,q)$, which define the covariance and contravariance of the tensor associated with that representation. But, if I'm not wrong, and, $n-1$ are needed to identify every irreducible representation of $mathrm{SU}(n)$. I have no proof of this but it makes sense if we look at the Young Tableaux method.
My questions, then, are:
Do tensor methods for finding irreducible representations of $mathrm{SU}(n)$ exist for $n > 3$?
If so, which are those?
If not, how is it that they exist for $n leq 3$, why do they work?
representation-theory lie-groups tensors
asked Jan 28 at 22:38
TeicDaunTeicDaun
31116
$endgroup$
$begingroup$
The irreducible representations are determined by their restriction to a maximal torus which has dimension $n-1$, these are the weights.
$endgroup$
– Charlie Frohman
Jan 29 at 2:27
1
$begingroup$
Yes, these are the Dynkin indices: the number of single boxes, two-box columns, 3-box columns...etc. You want a tutorial? Slansky 1981.
$endgroup$
– Cosmas Zachos
Feb 4 at 0:01
add a comment |
$begingroup$
I'm familiar with the Young Tableaux method for finding irreducible representations of $mathrm{SU}(n)$ for $ninmathbb{N}$, and I also know of the tensor methods for finding irreducible representations of $mathrm{SU}(3)$.
Tensor methods for $mathrm{SU}(3)$ seem to rely heavily on the fact that irreducible representations of this group are identified by two natural numbers $(p,q)$, which define the covariance and contravariance of the tensor associated with that representation. But, if I'm not wrong, and, $n-1$ are needed to identify every irreducible representation of $mathrm{SU}(n)$. I have no proof of this but it makes sense if we look at the Young Tableaux method.
My questions, then, are:
Do tensor methods for finding irreducible representations of $mathrm{SU}(n)$ exist for $n > 3$?
If so, which are those?
If not, how is it that they exist for $n leq 3$, why do they work?
representation-theory lie-groups tensors
asked Jan 28 at 22:38
TeicDaunTeicDaun
31116
$endgroup$
I'm familiar with the Young Tableaux method for finding irreducible representations of $mathrm{SU}(n)$ for $ninmathbb{N}$, and I also know of the tensor methods for finding irreducible representations of $mathrm{SU}(3)$.
Tensor methods for $mathrm{SU}(3)$ seem to rely heavily on the fact that irreducible representations of this group are identified by two natural numbers $(p,q)$, which define the covariance and contravariance of the tensor associated with that representation. But, if I'm not wrong, and, $n-1$ are needed to identify every irreducible representation of $mathrm{SU}(n)$. I have no proof of this but it makes sense if we look at the Young Tableaux method.
My questions, then, are:
Do tensor methods for finding irreducible representations of $mathrm{SU}(n)$ exist for $n > 3$?
If so, which are those?
If not, how is it that they exist for $n leq 3$, why do they work?
representation-theory lie-groups tensors
representation-theory lie-groups tensors
asked Jan 28 at 22:38
TeicDaunTeicDaun
31116
asked Jan 28 at 22:38
TeicDaunTeicDaun
31116
edited Feb 2 at 16:58
TeicDaun
asked Jan 28 at 22:38
TeicDaunTeicDaun
31116
asked Jan 28 at 22:38
TeicDaunTeicDaun
31116
asked Jan 28 at 22:38
TeicDaunTeicDaun
31116
31116
$begingroup$
The irreducible representations are determined by their restriction to a maximal torus which has dimension $n-1$, these are the weights.
$endgroup$
– Charlie Frohman
Jan 29 at 2:27
1
$begingroup$
Yes, these are the Dynkin indices: the number of single boxes, two-box columns, 3-box columns...etc. You want a tutorial? Slansky 1981.
$endgroup$
– Cosmas Zachos
Feb 4 at 0:01
add a comment |
$begingroup$
The irreducible representations are determined by their restriction to a maximal torus which has dimension $n-1$, these are the weights.
$endgroup$
– Charlie Frohman
Jan 29 at 2:27
1
$begingroup$
Yes, these are the Dynkin indices: the number of single boxes, two-box columns, 3-box columns...etc. You want a tutorial? Slansky 1981.
$endgroup$
– Cosmas Zachos
Feb 4 at 0:01
$begingroup$
The irreducible representations are determined by their restriction to a maximal torus which has dimension $n-1$, these are the weights.
$endgroup$
– Charlie Frohman
Jan 29 at 2:27
$begingroup$
The irreducible representations are determined by their restriction to a maximal torus which has dimension $n-1$, these are the weights.
$endgroup$
– Charlie Frohman
Jan 29 at 2:27
1
1
$begingroup$
Yes, these are the Dynkin indices: the number of single boxes, two-box columns, 3-box columns...etc. You want a tutorial? Slansky 1981.
$endgroup$
– Cosmas Zachos
Feb 4 at 0:01
$begingroup$
Yes, these are the Dynkin indices: the number of single boxes, two-box columns, 3-box columns...etc. You want a tutorial? Slansky 1981.
$endgroup$
– Cosmas Zachos
Feb 4 at 0:01
add a comment |
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$begingroup$
The irreducible representations are determined by their restriction to a maximal torus which has dimension $n-1$, these are the weights.
$endgroup$
– Charlie Frohman
Jan 29 at 2:27
1
$begingroup$
Yes, these are the Dynkin indices: the number of single boxes, two-box columns, 3-box columns...etc. You want a tutorial? Slansky 1981.
$endgroup$
– Cosmas Zachos
Feb 4 at 0:01