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.










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  • You are right. The statement in this form is not correct (you can construct small examples where it fails).
    – Tobias Kildetoft
    Nov 30 '18 at 12:28
















0














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.










share|cite|improve this question






















  • You are right. The statement in this form is not correct (you can construct small examples where it fails).
    – Tobias Kildetoft
    Nov 30 '18 at 12:28














0












0








0







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.










share|cite|improve this question













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






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asked Nov 30 '18 at 12:01









Luke MathwalkerLuke Mathwalker

352212




352212












  • You are right. The statement in this form is not correct (you can construct small examples where it fails).
    – Tobias Kildetoft
    Nov 30 '18 at 12:28


















  • You are right. The statement in this form is not correct (you can construct small examples where it fails).
    – Tobias Kildetoft
    Nov 30 '18 at 12:28
















You are right. The statement in this form is not correct (you can construct small examples where it fails).
– Tobias Kildetoft
Nov 30 '18 at 12:28




You are right. The statement in this form is not correct (you can construct small examples where it fails).
– Tobias Kildetoft
Nov 30 '18 at 12:28










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