$SUleft(Nright)$ Dynkin labels, how to compute
Let $V$ be somecomplex irreducible representation of $SUleft(Nright)$. I read that to compute the Dynkin labels of the weights, one can take the highest weight and then subtract from it the rows of the Cartan matrix. This makes sense, since weights of an irreducible representation must differ by roots.
My question is, which rows of the Cartan are the correct ones to use? For instance, for $SUleft(5right)$, suppose the highest weight is $left(0,0,0,1right)$. I know the Cartan matrix is
$$left(begin{array}{cccc}
2 & -1\
-1 & 2 & -1\
& -1 & 2 & -1\
& & -1 & 2
end{array}right)$$
Which rows do I subtract from the highest weight Dynkin label to obtain the other Dynkin labels, and when do I stop?
representation-theory lie-groups lie-algebras
asked Nov 28 '18 at 19:41
Joshua Tilley
549313
add a comment |
Let $V$ be somecomplex irreducible representation of $SUleft(Nright)$. I read that to compute the Dynkin labels of the weights, one can take the highest weight and then subtract from it the rows of the Cartan matrix. This makes sense, since weights of an irreducible representation must differ by roots.
My question is, which rows of the Cartan are the correct ones to use? For instance, for $SUleft(5right)$, suppose the highest weight is $left(0,0,0,1right)$. I know the Cartan matrix is
$$left(begin{array}{cccc}
2 & -1\
-1 & 2 & -1\
& -1 & 2 & -1\
& & -1 & 2
end{array}right)$$
Which rows do I subtract from the highest weight Dynkin label to obtain the other Dynkin labels, and when do I stop?
representation-theory lie-groups lie-algebras
asked Nov 28 '18 at 19:41
Joshua Tilley
549313
add a comment |
Let $V$ be somecomplex irreducible representation of $SUleft(Nright)$. I read that to compute the Dynkin labels of the weights, one can take the highest weight and then subtract from it the rows of the Cartan matrix. This makes sense, since weights of an irreducible representation must differ by roots.
My question is, which rows of the Cartan are the correct ones to use? For instance, for $SUleft(5right)$, suppose the highest weight is $left(0,0,0,1right)$. I know the Cartan matrix is
$$left(begin{array}{cccc}
2 & -1\
-1 & 2 & -1\
& -1 & 2 & -1\
& & -1 & 2
end{array}right)$$
Which rows do I subtract from the highest weight Dynkin label to obtain the other Dynkin labels, and when do I stop?
representation-theory lie-groups lie-algebras
asked Nov 28 '18 at 19:41
Joshua Tilley
549313
Let $V$ be somecomplex irreducible representation of $SUleft(Nright)$. I read that to compute the Dynkin labels of the weights, one can take the highest weight and then subtract from it the rows of the Cartan matrix. This makes sense, since weights of an irreducible representation must differ by roots.
My question is, which rows of the Cartan are the correct ones to use? For instance, for $SUleft(5right)$, suppose the highest weight is $left(0,0,0,1right)$. I know the Cartan matrix is
$$left(begin{array}{cccc}
2 & -1\
-1 & 2 & -1\
& -1 & 2 & -1\
& & -1 & 2
end{array}right)$$
Which rows do I subtract from the highest weight Dynkin label to obtain the other Dynkin labels, and when do I stop?
representation-theory lie-groups lie-algebras
representation-theory lie-groups lie-algebras
asked Nov 28 '18 at 19:41
Joshua Tilley
549313
asked Nov 28 '18 at 19:41
Joshua Tilley
549313
asked Nov 28 '18 at 19:41
Joshua Tilley
549313
asked Nov 28 '18 at 19:41
Joshua Tilley
549313
asked Nov 28 '18 at 19:41
Joshua Tilley
549313
549313
add a comment |
add a comment |
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