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\title{Math 3094 Humphreys Sec 1.3 Notes}
\author{E. Gunawan, University of Connecticut}
\date{Fall 2018}
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\section*{Proof of Theorem in Section 1.3 of Humphreys}

Let $\Pi$ be a positive system of a root system $\Phi$. Suppose $D$ is a minimal subset of $\Phi$ subject to the requirement that 
\begin{equation}
\label{eq:minimal_subset}
\text{each root in $\Pi$ is a nonnegative linear combination of $D$.}
\end{equation}
Prove that \[ \langle \alpha, \beta \rangle \leq 0 \text{ for all pairs } \alpha \neq \beta \in D.\]

How to do this problem: This inequality statement is labeled (1) on \cite[page 8]{Hum90}. The inequality is proven on \cite[page 9]{Hum90}. You can also read \cite[page 34]{Der14}.

\begin{proof}
Suppose $\langle \beta, \alpha \rangle > 0$ for some $\alpha \neq \beta \in D$. Then the formula for a reflection gives 
\[
s_\alpha(\beta) = \beta - c\alpha, ~ 
\text{ 
with $c=2 \frac{\langle \beta, \alpha \rangle}{\langle \alpha, \alpha \rangle} > 0$.}
\]
To see this, recall that $\langle \alpha, \alpha \rangle > 0$ because $\langle , \rangle$ is an inner product (\textit{i.e.}, positive definite symmetric bilinear form) on $\mathbb{R}^n$. Since $s_\alpha(\beta) \in \Phi = \Pi \cup -\Pi$, either $s_\alpha(\beta)\in \Pi$ or $-s_\alpha(\beta) \in \Pi$. The argument that the former case is impossible is done in \cite[Section 1.3]{Hum90}, and we will now show that the latter case is also impossible.

Suppose $-s_\alpha(\beta) \in \Pi$. Then 
\begin{align*}
-s_\alpha(\beta) &= \sum_{\gamma\in\Delta} c_\gamma \gamma 
\text{ with $c_\gamma \geq 0$ by definition of $D$}\\
&=c_\alpha \alpha + \sum_{\gamma \in \Delta, \gamma \neq \alpha} c_\gamma \gamma.
\end{align*}
Hence 
\begin{equation}
\label{eq:negative_of_s_alpha_beta}
-s_\alpha(\beta) = -\beta + c\alpha =  c_\alpha \alpha + \sum_{\gamma \in \Delta, \gamma \neq \alpha} c_\gamma \gamma.
\end{equation}

In case $c_{\alpha} < c$, equation (\ref{eq:negative_of_s_alpha_beta}) implies  
\[
(c-c_\alpha) \alpha = \beta + \sum_{\gamma \in \Delta, \gamma \neq \alpha} c_\gamma \gamma,
\]
which is a non-negative linear combination of $D \backslash \{ \alpha \}$.
Since $0 < c - c_{\alpha}$, this allows us to discard $\alpha$ from the subset $D$ and still have a subset satisfying the requirement (\ref{eq:minimal_subset}), contradicting the minimality of $D$.

In case  $c < c_{\alpha}$, equation (\ref{eq:negative_of_s_alpha_beta}) gives 
\[
0 = (c_\alpha - c) \alpha + \beta + \sum_{\gamma \in \Delta, \gamma \neq \alpha} c_\gamma \gamma,
\]
which is a non-negative linear combination of $D$. But all these roots are positive and the coefficient for $\beta$ has a positive coefficient (at least $1$), so the right hand side cannot equal to $0$ (by definition of total ordering). 

We've shown that $-s_\alpha(\beta) \notin \Pi$. Since $s_\alpha(\beta) \notin \Pi$ as shown in \cite[Section 1.3]{Hum90}, we have $\alpha(\beta) \notin \Phi$, which is a contradiction. 
\end{proof}

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