# Bases $\Delta$ for $\Phi$ (Friday, October 14)


:::{.remark}
From a previous discussion: given a rank $n$ root system $\Phi$ with $n\geq 2$, is $\RR\gens{ \alpha, \beta} \intersect \Phi$ always a rank 2 root system?
The answer is yes! This follows readily from just checking the axioms directly.

:::

:::{.remark}
For a regular \( \gamma \in \EE\sm \Union_{\alpha\in \Phi} P_ \alpha \), define $\Phi^+(\gamma) \da \ts{ \beta\in \Phi \st (\beta, \gamma) > 0}$ and let $\Delta( \gamma)$ be the indecomposable elements of $\Phi^+( \gamma)$:

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:::{.theorem title="?"}
$\Delta( \gamma)$ is a base for $\Phi$, and every base is of this form.
:::

:::{.definition title="?"}
The connected components of $\EE \sm \Union_{ \alpha\in \Phi} P_ \alpha$ are called the **(open) Weyl chambers** of $\EE$.
Each regular $\gamma$ belongs to some Weyl chamber, which we'll denote $C(\gamma)$.
:::

:::{.remark}
Note that $C(\gamma) = C(\gamma') \iff \gamma, \gamma'$ are on the same side of every root hyperplane $P_\alpha$ for $\alpha\in \Phi$, which happens iff $\Phi^+( \gamma) = \Phi^+(\gamma;) \iff \Delta( \gamma) = \Delta(\gamma')$, so there is a bijection
\[
\correspond{
  \text{Weyl chambers}
}
&\mapstofrom
\correspond{
  \text{Bases for $\Phi$}
}
.\]
Note also that $W$ sends one Weyl chamber to another: any $s_ \alpha$ preserves the connected components $\EE\sm \Union_{ \beta\in \Phi} P_{\beta}$, so if $\gamma$ is regular and $\sigma\in W$ and $\sigma( \gamma) = \gamma'$ for some regular $\gamma'$, then $\sigma(C(\gamma)) = C(\gamma')$.
$W$ also acts on bases for $\Phi$: if $\Delta$ is a base for $\Phi$, then $\sigma( \Delta)$ is still a basis for $\EE$ since $\sigma$ is an invertible linear transformation.
Since $\sigma( \Phi) = \Phi$ by the axioms, any root $\alpha\in \Phi$ is of the form $\sigma(\beta)$ for some \( \beta\in \Phi \), but writing $\beta = \sum _{\alpha\in\Delta} k_ \alpha \alpha$ with all $k_\alpha$ the same sign, $\sigma( \beta) = \sum_{ \alpha\in \Delta} k_ \alpha \sigma( \alpha)$ is a linear combination of elements in $\sigma(\Delta)$ with coefficients of the same sign.

The actions of $W$ on chambers and bases are compatible: if $\Delta = \Delta( \gamma)$ then \( \sigma( \Delta) = \Delta( \sigma( \gamma)) \), since $\sigma(\Phi^+( \gamma)) = \Phi^+( \sigma( \gamma))$ since $W \leq \Orth(\EE)$ and thus $( \sigma \gamma, \sigma \alpha) = (\gamma, \alpha)$.
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:::{.lemma title="A"}
Fix a base $\Delta \subset \Phi$, which decomposes $\Phi = \Phi^+ \disjoint \Phi^-$.
If \( \beta\in \Phi^+ \sm \Delta \), then \( \beta- \alpha\in \Phi^+ \) for some $\alpha\in\Delta$.
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:::{.proof title="?"}
If $( \beta, \alpha)\leq 0$ for all $\alpha\in \Delta$, then the proof of theorem 10.1 would show $\beta = 0$ by taking $\eps \da \beta$ in step 3.
One can then find an $\alpha\in \Delta$ with $( \beta, \alpha) > 0$, where clearly $\beta \neq \pm \alpha$.
By lemma 9.4, $\beta- \alpha\in \Phi$.
Why is this positive?
Note that $\beta$ has at least one coefficient for a simple root (not the coefficient for $\alpha$) which is strictly positive, and thus all coefficients are $\geq 0$.
This coefficient stays the same in $\beta- \alpha$, so its coefficients are all non-negative by axiom B2 and $\beta-\alpha\in \Phi^+$.
:::

:::{.corollary title="?"}
Each $\beta\in \Phi^+$ can be written as \( \alpha_1 + \cdots + \alpha_k \) where $\alpha_i\in \Delta$ not necessarily distinct, such that each truncated sum \( \alpha_1 + \cdots + \alpha_i \) for $1\leq i \leq k$ is a positive root.
One proves this by induction on the height of $\beta$.
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:::{.lemma title="B"}
Let $\alpha \in \Delta$, then $s_ \alpha$ permutes $\Phi^+ \smts{ \alpha}$.
:::

:::{.proof title="?"}
Let $\beta \in \Phi^+\smts{\alpha}$; if $\beta = \sum _{\gamma\in \Delta} k_ \gamma \gamma$ with $k_ \gamma \in \ZZ_{\geq 0}$, since $\beta\neq \alpha$, some $k_ \gamma > 0$ for some $\gamma \neq \alpha$.
Using the formula $s_ \alpha( \beta) = \beta- \inp \beta \alpha \alpha$ still has coefficient $k_ \gamma$ for $\gamma$.
Thus $s_ \alpha( \beta) \in \Phi^+$ and $s_ \alpha( \beta)\neq \alpha$ since $s_ \alpha(- \alpha) = \alpha$ and $s_\alpha$ is bijective, and so $s_\alpha(\beta)\in \Phi^+\smts{ \alpha}$.
As a result, $s_\alpha$ permutes this set since it is invertible.
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:::{.corollary title="?"}
Let 
\[
\rho \da {1\over 2}\sum_{ \beta\in \Phi^+} \beta \quad\in \EE
.\]
Then $s_\alpha( \rho) = \rho- \alpha$ for all \( \alpha\in \Delta \), and $s_ \alpha( \rho) = \rho$.

> Note that Humphreys uses $\delta$, but nobody uses this notation.

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:::{.lemma title="C, The Deletion Condition"}
Let \( \alpha_1,\cdots \alpha_t \in \Delta \) be not necessarily distinct simple roots, and write $s_i \da s_{\alpha_i}$.
If $s_1 \cdots s_{t-1}(\alpha_t) < 0$, then for some $1\leq u \leq t$ one has
\[
s_1\cdots s_t = s_1\cdots s_{u-1} s_{u+1} s_{t-1}
,\]
so one can delete $s_u$ and $s_t$ to get a shorter product of reflections.
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:::{.proof title="?"}
For $0\leq i\leq t-1$, let $\beta_i \da s_{i+1} \cdots s_{t-1}( \alpha_t)$ and $\beta_{t-1} \da \alpha_t$.
Since $\beta_0 \da s_1\cdots s_{t-1}( \alpha_t ) < 0$ and $\beta_{t-1} = \alpha_t > 0$, there must be a smallest index $u$ such that $\beta_u > 0$.
Note that $u\geq 1$ since $\beta_0$ is negative.
Then 
\[
s_u( \beta_u) 
&= s_u s_{u+1} \cdots s_{t-1}( \alpha_t) \\
&= \beta_{u-1} \\
& < 0
\]
by choice of $u$.
Noting that $\beta_u = s_{u+1}\cdots s_{t-1} (\alpha_t)$, by lemma B, $s_u = s_{\alpha_u}$ permutes roots other than $\alpha_u$ since $\beta_u > 0$ and $s_{\alpha_u}( \beta_u) < 0$.
By lemma 9.2, write
\[
s_{\alpha_u} = s_{\beta_u} = s_{ s_{u+1}\cdots s_{t-1}( \alpha_t ) } = (s_{u+1} \cdots s_{t-1}) s_{\alpha_u} (s_{u+1}\cdots s_{t-1})\inv
.\]
Multiply both sides on the left by $(s_1\cdots s_u)$ and on the right by $(s_{u+1}\cdots s_{t-1})$ to obtain
\[
(s_1 \cdots s_{u-1})(s_{u+1}\cdots s_{t-1}) = (s_1\cdots s_u)(s_{u+1}\cdots s_t), \qquad s_t \da s_{\alpha_t}
.\]

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:::{.corollary title="?"}
If $\sigma = s_1\cdots s_t$ is an expression for $w\in W$ in terms of simple reflections (which we don't yet know exists, but it does) with $t$ minimal, then $\sigma( \alpha_t) < 0$.
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