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Aug
13
revised Does there exist $\gamma \in \Phi^+$ such that $(\gamma,\alpha)$ and $(\gamma,\beta)$ are non-zero?
added 99 characters in body
Aug
13
revised Does there exist $\gamma \in \Phi^+$ such that $(\gamma,\alpha)$ and $(\gamma,\beta)$ are non-zero?
added 50 characters in body
Aug
13
revised Does there exist $\gamma \in \Phi^+$ such that $(\gamma,\alpha)$ and $(\gamma,\beta)$ are non-zero?
added 444 characters in body
Aug
11
awarded  Disciplined
Aug
9
asked Does there exist $\gamma \in \Phi^+$ such that $(\gamma,\alpha)$ and $(\gamma,\beta)$ are non-zero?
Jul
8
comment Singular points of evolute and pedal
Very kind answer. Thank you @IvoTerek.
Jul
8
accepted Singular points of evolute and pedal
Jul
8
comment Singular points of evolute and pedal
I think we should prove that $A=0 \iff \gamma=0$. But haven't gotten any clue yet.
Jul
8
comment Singular points of evolute and pedal
I found $ P' = -kA$ where $A=(\gamma T)N +(\gamma N) T$. To solve $A=0$, using the facts that $NT=0, T^2=1$ gives us $\gamma N=0$. Differentiating both sides we get $-k\gamma'\gamma=0$ i.e. $k(\|\gamma\|^2)'=0$. Since $\|\gamma\|^2$ is constant then this equation is always true?! Does this make any sense?
Jul
8
asked Singular points of evolute and pedal
Jul
1
comment Greatest elements in crystallographic root systems
One more question has come up in my mind. Assume that you're considering root systems in general but not $G_2$. Can you always suppose that long roots have length 2 and short roots have length 1?
Jun
27
comment Greatest elements in crystallographic root systems
$a'$ and $b'$ are actually simple roots?
Jun
27
comment Greatest elements in crystallographic root systems
What did you mean by orthogonal action?
Jun
27
comment Greatest elements in crystallographic root systems
Thank you @Sebastian. Can you give sources where you got your above statement? For instance some key words, Weyl group acts transitively, orthogonally, there are long roots $a'$ and short root $b'$ which is their inner product is 1...
Jun
26
comment Greatest elements in crystallographic root systems
How did you get the statement "By transitivity there exists a long root a such that (a,b)=-1"?
Jun
25
accepted Coefficients of positive roots in term of simple roots
Jun
24
comment Equivalent definitions of positive root system
Do you have any material to prove your definition of positive root systems at the beginning?
Jun
23
asked Coefficients of positive roots in term of simple roots
Jun
23
awarded  Enthusiast
Jun
22
comment A general pattern to find the roots of the classical lie algebras
Thank you for very nice answer. However did your observation have any proof? I see a quick corollary that there always exists a number 1 in every sequence of coefficients. Do you have any quick explanation for my statement?