What is the weight system for these $suleft(5right)$ representations?
I need to work out the weight systems for the fundamental representation $mathbf{5}$ and the conjugate representation $overline{mathbf{5}}$. I'm not clear what this means. The 5 representation is of course just the representation of $suleft(5right)$ by itself. After picking a Cartan subalgebra as the diagonal matrices with zero trace, we can of course see that the roots are $L_i−L_j$ where $L_i$ picks out the ith element on the diagonal, and the weights are simply $L_i$ in this case. So what is the 'weight system'?
It is supposed to be the case that I can use the weight systems of representations to show for instance that $mathbf{5}otimes mathbf{5}=mathbf{5}oplus mathbf{15}$.
representation-theory lie-groups lie-algebras
asked Nov 28 at 3:06
Joshua Tilley
539313
add a comment |
I need to work out the weight systems for the fundamental representation $mathbf{5}$ and the conjugate representation $overline{mathbf{5}}$. I'm not clear what this means. The 5 representation is of course just the representation of $suleft(5right)$ by itself. After picking a Cartan subalgebra as the diagonal matrices with zero trace, we can of course see that the roots are $L_i−L_j$ where $L_i$ picks out the ith element on the diagonal, and the weights are simply $L_i$ in this case. So what is the 'weight system'?
It is supposed to be the case that I can use the weight systems of representations to show for instance that $mathbf{5}otimes mathbf{5}=mathbf{5}oplus mathbf{15}$.
representation-theory lie-groups lie-algebras
asked Nov 28 at 3:06
Joshua Tilley
539313
The representation labelled by $5$ is not the algebra itself, since that is way more than $5$-dimensional. It is instead the "defining" representation coming from this being a subalgebra of $mathfrak{gl}(5)$. In general, labelling the irreducible representations by their dimensions here is going to cause a lot of issues, since there will potentially be many representations of a given dimension.
– Tobias Kildetoft
Nov 28 at 9:47
5 ⊗ 5 = 10 ⊕ 15.
– Cosmas Zachos
Dec 18 at 1:38
add a comment |
I need to work out the weight systems for the fundamental representation $mathbf{5}$ and the conjugate representation $overline{mathbf{5}}$. I'm not clear what this means. The 5 representation is of course just the representation of $suleft(5right)$ by itself. After picking a Cartan subalgebra as the diagonal matrices with zero trace, we can of course see that the roots are $L_i−L_j$ where $L_i$ picks out the ith element on the diagonal, and the weights are simply $L_i$ in this case. So what is the 'weight system'?
It is supposed to be the case that I can use the weight systems of representations to show for instance that $mathbf{5}otimes mathbf{5}=mathbf{5}oplus mathbf{15}$.
representation-theory lie-groups lie-algebras
asked Nov 28 at 3:06
Joshua Tilley
539313
I need to work out the weight systems for the fundamental representation $mathbf{5}$ and the conjugate representation $overline{mathbf{5}}$. I'm not clear what this means. The 5 representation is of course just the representation of $suleft(5right)$ by itself. After picking a Cartan subalgebra as the diagonal matrices with zero trace, we can of course see that the roots are $L_i−L_j$ where $L_i$ picks out the ith element on the diagonal, and the weights are simply $L_i$ in this case. So what is the 'weight system'?
It is supposed to be the case that I can use the weight systems of representations to show for instance that $mathbf{5}otimes mathbf{5}=mathbf{5}oplus mathbf{15}$.
representation-theory lie-groups lie-algebras
representation-theory lie-groups lie-algebras
asked Nov 28 at 3:06
Joshua Tilley
539313
asked Nov 28 at 3:06
Joshua Tilley
539313
asked Nov 28 at 3:06
Joshua Tilley
539313
asked Nov 28 at 3:06
Joshua Tilley
539313
asked Nov 28 at 3:06
Joshua Tilley
539313
539313
The representation labelled by $5$ is not the algebra itself, since that is way more than $5$-dimensional. It is instead the "defining" representation coming from this being a subalgebra of $mathfrak{gl}(5)$. In general, labelling the irreducible representations by their dimensions here is going to cause a lot of issues, since there will potentially be many representations of a given dimension.
– Tobias Kildetoft
Nov 28 at 9:47
5 ⊗ 5 = 10 ⊕ 15.
– Cosmas Zachos
Dec 18 at 1:38
add a comment |
The representation labelled by $5$ is not the algebra itself, since that is way more than $5$-dimensional. It is instead the "defining" representation coming from this being a subalgebra of $mathfrak{gl}(5)$. In general, labelling the irreducible representations by their dimensions here is going to cause a lot of issues, since there will potentially be many representations of a given dimension.
– Tobias Kildetoft
Nov 28 at 9:47
5 ⊗ 5 = 10 ⊕ 15.
– Cosmas Zachos
Dec 18 at 1:38
The representation labelled by $5$ is not the algebra itself, since that is way more than $5$-dimensional. It is instead the "defining" representation coming from this being a subalgebra of $mathfrak{gl}(5)$. In general, labelling the irreducible representations by their dimensions here is going to cause a lot of issues, since there will potentially be many representations of a given dimension.
– Tobias Kildetoft
Nov 28 at 9:47
The representation labelled by $5$ is not the algebra itself, since that is way more than $5$-dimensional. It is instead the "defining" representation coming from this being a subalgebra of $mathfrak{gl}(5)$. In general, labelling the irreducible representations by their dimensions here is going to cause a lot of issues, since there will potentially be many representations of a given dimension.
– Tobias Kildetoft
Nov 28 at 9:47
5 ⊗ 5 = 10 ⊕ 15.
– Cosmas Zachos
Dec 18 at 1:38
5 ⊗ 5 = 10 ⊕ 15.
– Cosmas Zachos
Dec 18 at 1:38
add a comment |
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The representation labelled by $5$ is not the algebra itself, since that is way more than $5$-dimensional. It is instead the "defining" representation coming from this being a subalgebra of $mathfrak{gl}(5)$. In general, labelling the irreducible representations by their dimensions here is going to cause a lot of issues, since there will potentially be many representations of a given dimension.
– Tobias Kildetoft
Nov 28 at 9:47
5 ⊗ 5 = 10 ⊕ 15.
– Cosmas Zachos
Dec 18 at 1:38