Dominant weights in root system
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Suppose a root system is given. Let $gamma$ be an integral, dominant weight and $rho$ be the half sum of positive roots. I have been told that one can choose an integral, dominant weight $mu$ such that $gamma-mu=rho$. To me it is only clear that $mu=gamma-rho$ is integral, since both summands are integral. However I don't believe in the dominance of $mu$ by the minus sign in front of $rho$. Am I overseeing something? Thanks in advance.
lie-algebras root-systems weyl-group
asked Nov 30 '18 at 12:01
Luke MathwalkerLuke Mathwalker
352212
$endgroup$
add a comment |
$begingroup$
Suppose a root system is given. Let $gamma$ be an integral, dominant weight and $rho$ be the half sum of positive roots. I have been told that one can choose an integral, dominant weight $mu$ such that $gamma-mu=rho$. To me it is only clear that $mu=gamma-rho$ is integral, since both summands are integral. However I don't believe in the dominance of $mu$ by the minus sign in front of $rho$. Am I overseeing something? Thanks in advance.
lie-algebras root-systems weyl-group
asked Nov 30 '18 at 12:01
Luke MathwalkerLuke Mathwalker
352212
$endgroup$
$begingroup$
You are right. The statement in this form is not correct (you can construct small examples where it fails).
$endgroup$
– Tobias Kildetoft
Nov 30 '18 at 12:28
add a comment |
$begingroup$
Suppose a root system is given. Let $gamma$ be an integral, dominant weight and $rho$ be the half sum of positive roots. I have been told that one can choose an integral, dominant weight $mu$ such that $gamma-mu=rho$. To me it is only clear that $mu=gamma-rho$ is integral, since both summands are integral. However I don't believe in the dominance of $mu$ by the minus sign in front of $rho$. Am I overseeing something? Thanks in advance.
lie-algebras root-systems weyl-group
asked Nov 30 '18 at 12:01
Luke MathwalkerLuke Mathwalker
352212
$endgroup$
Suppose a root system is given. Let $gamma$ be an integral, dominant weight and $rho$ be the half sum of positive roots. I have been told that one can choose an integral, dominant weight $mu$ such that $gamma-mu=rho$. To me it is only clear that $mu=gamma-rho$ is integral, since both summands are integral. However I don't believe in the dominance of $mu$ by the minus sign in front of $rho$. Am I overseeing something? Thanks in advance.
lie-algebras root-systems weyl-group
lie-algebras root-systems weyl-group
asked Nov 30 '18 at 12:01
Luke MathwalkerLuke Mathwalker
352212
asked Nov 30 '18 at 12:01
Luke MathwalkerLuke Mathwalker
352212
asked Nov 30 '18 at 12:01
Luke MathwalkerLuke Mathwalker
352212
asked Nov 30 '18 at 12:01
Luke MathwalkerLuke Mathwalker
352212
asked Nov 30 '18 at 12:01
Luke MathwalkerLuke Mathwalker
352212
352212
$begingroup$
You are right. The statement in this form is not correct (you can construct small examples where it fails).
$endgroup$
– Tobias Kildetoft
Nov 30 '18 at 12:28
add a comment |
$begingroup$
You are right. The statement in this form is not correct (you can construct small examples where it fails).
$endgroup$
– Tobias Kildetoft
Nov 30 '18 at 12:28
$begingroup$
You are right. The statement in this form is not correct (you can construct small examples where it fails).
$endgroup$
– Tobias Kildetoft
Nov 30 '18 at 12:28
$begingroup$
You are right. The statement in this form is not correct (you can construct small examples where it fails).
$endgroup$
– Tobias Kildetoft
Nov 30 '18 at 12:28
add a comment |
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$begingroup$
You are right. The statement in this form is not correct (you can construct small examples where it fails).
$endgroup$
– Tobias Kildetoft
Nov 30 '18 at 12:28