Deletion and Exchange Conditions

Length Function

For any $w\in W$, $w$ can be written as a product of simple reflections, $w=s_1\cdots s_r$ where $s_i=s_{{\alpha }_i}$ for some ${\alpha }_i\in \Delta$. The length $l(w)$ of $w$ is defined to be the smallest $r$ for which such an expression exists, and call the expression reduced. By convention, $l(1)=0$.

$n(\omega )$

Having fixed $\Delta$ and the corresponding positive system $\Pi$, define $n(\omega )$, number of positive roots sent to negative roots by $\omega$ as follow $n(\omega )=\text{Card}(\Pi \cap w^{-1}(-\Pi ))$

Lemma

Let $\alpha \in \Delta$, $\omega \in W$. Then: (a) $\omega \alpha >0⟹n(\omega s_{\alpha })=n(\omega )+1$ (b) $\omega \alpha <0⟹n(\omega s_{\alpha })=n(\omega )-1$ (c) ${\omega }^{-1}\alpha >0⟹n(s_{\alpha }\omega )=n(\omega )+1$ (d)${\omega }^{-1}\alpha <0⟹n(s_{\alpha }\omega )=n(\omega )-1$

Proof (a) $\omega \alpha >0$, $\Pi \cap (\omega s_{\alpha }{)}^{-1}(-\Pi )=\alpha \bigsqcup \Pi \cap {\omega }^{-1}(-\Pi ),$ $\omega s_{\alpha }\alpha =-\omega \alpha \in -\Pi$, so $\alpha \in \Pi \cap (\omega s_{\alpha }{)}^{-1}(-\Pi )$, $\omega \alpha >0$, which means $\omega$ doesn’t send $\alpha$ to negative, so $\alpha \notin (\Pi \cap {\omega }^{-1}(-\Pi ))$, together with $s_{\alpha }(\Pi \setminus \{\alpha \})=\Pi \setminus \{\alpha \}$, we get above equality. Then $n(\omega s_{\alpha })=n(\omega )+1$ (b) $\omega \alpha <0$, $\Pi \cap (\omega s_{\alpha }{)}^{-1}(-\Pi )=\Pi \cap {\omega }^{-1}(-\Pi )\setminus \{\alpha \},$ $\omega s_{\alpha }\alpha =-\omega \alpha \in \Pi$, so $\alpha \notin \Pi \cap (\omega s_{\alpha }{)}^{-1}(-\Pi )$, together with $s_{\alpha }(\Pi \setminus \{\alpha \})=\Pi \setminus \{\alpha \}$, the set reflects by $\omega$ would also reflect by $\omega s_{\alpha }$, thus we get above equality. Then $n(\omega s_{\alpha })=n(\omega )-1$. (c) and (d), we replace $\omega$ by ${\omega }^{-1}$ and use the fact $n({\omega }^{-1}{s}_{\alpha })=n(s_{\alpha }\omega )$

Corollary

If $w\in W$ is written in any way as a product of simple reflections, say $w=s_1\cdots s_r$, then $n(w)\le r$. In particular, $n(w)\le l(w)$.

Deletion Condition

Fix a simple system $\Delta$. Let $w=s_1\cdots s_r$ be any expression of $w\in W$ as a product of simple reflections (say $s_i=s_{{\alpha }_i}$, with repetitions permitted). Suppose $n(w)<r$. Then there exist indices $1\le i<j\le r$ satisfying: (a) ${\alpha }_i=(s_{i+1}\cdots s_{j-1}){\alpha }_j$ (b) $s_{i+1}{s}_{i+2}\cdots s_j=s_i{s}_{i+1}\cdots s_{j-1}$ (c) $w=s_1\cdots \widehat{s_i}\cdots \widehat{s_j}\cdots s_r$

Proof (a) If $s_1\cdots s_{k-1}{\alpha }_k>0$, by (a) in above lemma, $n(s_1\cdots s_{k-1}{s}_k)=n(s_1\cdots s_{k-1})+1$, then if for any $k$, $s_1\cdots s_{k-1}{\alpha }_k>0$, $n(w)=r$, contradicts to $n(w)<r$. So there exists $j$ that $s_1\cdots s_{j-1}{\alpha }_j<0$. Since ${\alpha }_j>0$, there exist $i$ that $s_i(s_{i+1}\cdots s_{j-1}{\alpha }_j)>0$ and $s_{i+1}\cdots s_{j-1}{\alpha }_j<0$. From proposition $s_i(\Pi \setminus \{{\alpha }_i\})=\Pi \setminus \{{\alpha }_i\}$, ${\alpha }_i=(s_{i+1}\cdots s_{j-1}){\alpha }_j$. (b) From previous proposition, $w{s}_{{\alpha }_i}{w}^{-1}=s_{w{\alpha }_i}$, so $(s_{i+1}\cdots s_{j-1})s_j(s_{j-1}\cdots s_{i+1})=s_{{\alpha }_i}=s_i,$ rearrange we get $s_{i+1}{s}_{i+2}\cdots s_j=s_i{s}_{i+1}\cdots s_{j-1}$. (c) Apply $s_i$ on both side of equality in $(b)$, $s_i(s_{i+1}{s}_{i+2}\cdots s_j)=s_{i+1}\cdots s_{j-1},$ substitute it in the original expression of $w$, $w=s_1\cdots \widehat{s_i}\cdots \widehat{s_j}\cdots s_r$

Corollary

If $w\in W$, then $n(w)=l(w)$.

Proof We get $n(w)\le l(w)$, from previous corollary, if $n(w)<l(w)$, consider $w=s_1\cdots s_r$ a reduced expression, $l(w)=r$, $n(w)<r$, by above theorem (c), then we can further reduce $w$ to length $r-2$, contradiction, so $n(w)=r$.

Exchange Condition

Let $w=s_1\cdots s_r$ (not necessarily reduced), where each $s_i$ is a simple reflection. If $\ell (ws)<\ell (w)$ for some simple reflection $s=s_{\alpha }$, then there exists an index $i$ for which $ws=s_1\cdots {\hat{s}}_i\cdots s_r$ (and thus $w=s_1\cdots {\hat{s}}_i\cdots s_{r}s$, with a factor $s$ exchanged for a factor $s_i$). In particular, $w$ has a reduced expression ending in $s$ if and only if $\ell (ws)<\ell (w)$.

Theorem

Let $\Delta$ be a simple system, $\Pi$ the corresponding positive system. The following conditions on $w\in W$ are equivalent: (a) $w\Pi =\Pi$; (b) $w\Delta =\Delta$; (c) $n(w)=0$; (d) $\ell (w)=0$; (e) $w=1.$