A transported simple root lies in the positive span of the simple root and the inversion roots

We prove the following statement $P(\mathrm{\ell })$ for every $\mathrm{\ell }\ge 0$: for every $u\in W$ with $\mathrm{\ell }(u)=\mathrm{\ell }$, every $t\mathrm{\notin }S(u)$ and every reduced expression $u=s_1\cdots s_{\mathrm{\ell }}$ with prefix roots ${\gamma }_i:=\rho (s_1\cdots s_{i-1})e_{s_i}$, one has $\rho (u)e_t=e_t+{\sum }_{i=1}^{\mathrm{\ell }}{c}_i{\gamma }_i$ with $c_i\ge 0$ for all $i$. For the given pair $(w,s)$ we have $\mathrm{\ell }(w)=k$ by the reducedness of $w=r_1\cdots r_k$, so clause (1) is the instance $u=w$, $t=s$ of $P(k)$, clause (3) is the same statement quantified over all pairs $(u,s)$ with $s\mathrm{\notin }S(u)$, and clause (2) is obtained from it in the last steps.

Base case $\mathrm{\ell }=0$: here $u=1$, the expression is empty, and the homomorphism property in [F2] gives $\rho (1)e_t=e_t$, which is the claimed identity with the empty sum.

Induction hypothesis: assume $P(m)$ for all $m\le \mathrm{\ell }-1$ and let $u=r_1{u}^{\mathrm{\prime }}$ be a reduced expression with $u^{\mathrm{\prime }}=r_2\cdots r_{\mathrm{\ell }}$, so $\mathrm{\ell }(u^{\mathrm{\prime }})=\mathrm{\ell }-1$ and $u^{\mathrm{\prime }}$ inherits $t\mathrm{\notin }S(u^{\mathrm{\prime }})$ from $t\mathrm{\notin }S(u)=\{r_1\}\cup S(u^{\mathrm{\prime }})$, using the support clause in F6.

Applying the hypothesis of step 1.3 to the pair $(u^{\mathrm{\prime }},t)$ and the reduced expression $r_2\cdots r_{\mathrm{\ell }}$ gives $\rho (u^{\mathrm{\prime }})e_t=e_t+{\sum }_{i=2}^{\mathrm{\ell }}{c}_i{\gamma }_i^{\mathrm{\prime }}$ with $c_i\ge 0$ and ${\gamma }_i^{\mathrm{\prime }}:=\rho (r_2\cdots r_{i-1})e_{r_i}$ the pairwise distinct prefix roots of $u^{\mathrm{\prime }}$; by the prefix-root formula [F5], $N((u^{\mathrm{\prime }}{)}^{-1})=\{{\gamma }_2^{\mathrm{\prime }},\dots ,{\gamma }_{\mathrm{\ell }}^{\mathrm{\prime }}\}$.

By [F2], $\rho (r_1)=r_{e_{r_1}}$; applying the reflection formula and Coxeter-form entries from [F1] gives $\rho (r_1)e_t=e_t-2B(e_t,e_{r_1})e_{r_1}=e_t+c_1{e}_{r_1}$ with $c_1:=-2B(e_t,e_{r_1})\ge 0$: indeed $t\mathrm{\ne }{r}_1$ because $t\mathrm{\notin }S(u)$ and $r_1\in S(u)$, and every off-diagonal value of $B$ is $\le 0$ since $m(s,t)\ge 2$ for $s\mathrm{\ne }t$ gives $\cos (\pi \mathrm{/}m(s,t))\ge 0$. Moreover $e_{r_1}={\gamma }_1$ is the first prefix root of $u$.

No prefix root ${\gamma }_i^{\mathrm{\prime }}$ of $u^{\mathrm{\prime }}$ equals $e_{r_1}$: if ${\gamma }_i^{\mathrm{\prime }}=e_{r_1}$, then $e_{r_1}\in N((u^{\mathrm{\prime }}{)}^{-1})$ by step 2.1; the inversion-set definition [F4] gives $\rho ((u^{\mathrm{\prime }}{)}^{-1})e_{r_1}\in {\mathrm{\Phi }}_{-}$. The root-length criterion [F7] then gives $\mathrm{\ell }((u^{\mathrm{\prime }}{)}^{-1}{r}_1)<\mathrm{\ell }((u^{\mathrm{\prime }}{)}^{-1})$, which by [F9] is $\mathrm{\ell }(r_1{u}^{\mathrm{\prime }})<\mathrm{\ell }(u^{\mathrm{\prime }})$, contradicting the reducedness of $u=r_1{u}^{\mathrm{\prime }}$.

For $i\ge 2$ we have ${\gamma }_i^{\mathrm{\prime }}\in {\mathrm{\Phi }}_{+}\setminus \{e_{r_1}\}$ by steps 2.1 and 3.1, so the simple-reflection action in F3 gives $\rho (r_1){\gamma }_i^{\mathrm{\prime }}\in {\mathrm{\Phi }}_{+}\setminus \{e_{r_1}\}$; and $\rho (r_1){\gamma }_i^{\mathrm{\prime }}=\rho (r_1\cdots r_{i-1})e_{r_i}={\gamma }_i$ is the $i$-th prefix root of $u$.

The homomorphism property in [F2] gives $\rho (u)=\rho (r_1)\rho (u^{\mathrm{\prime }})$, so steps 2.1, 2.2 and 4.1 give $\rho (u)e_t=e_t+c_1{\gamma }_1+{\sum }_{i=2}^{\mathrm{\ell }}{c}_i{\gamma }_i$ with all coefficients $\ge 0$; this is $P(\mathrm{\ell })$.

The base case of step 1.2 and the induction step, using step 1.3 to set up and step 5.1 to prove the inductive case, establish $P(\mathrm{\ell })$ for every $\mathrm{\ell }\ge 0$. In particular $P(k)$ gives, for the given pair $(w,s)$ and the reduced expression $w=r_1\cdots r_k$, the expansion $\rho (w)e_s=e_s+{\sum }_{i=1}^k{c}_i{\beta }_i$ with $c_i\ge 0$ for all $i$.

By the support clause F6, each prefix root ${\beta }_i=\rho (r_1\cdots r_{i-1})e_{r_i}$ has $r_1\cdots r_{i-1}\in W_{S(w)}$ and $r_i\in S(w)$; hence the parabolic root identity F8 gives ${\beta }_i\in {\mathrm{\Phi }}_{S(w)}=\mathrm{\Phi }\cap V_{S(w)}$. By the sign split F3, each ${\beta }_i\in {\mathrm{\Phi }}_{+}\subseteq V_{+}$ is a nonnegative combination of the simple roots $e_r$ with $r\in S(w)$ and has no $e_s$-coordinate, since $s\mathrm{\notin }S(w)$. Therefore every simple coordinate of $\rho (w)e_s$ is $\ge 0$ by step 6.1, its $e_s$-coordinate equals $1$, and its support satisfies ${\mathrm{supp}}_S(\rho (w)e_s)\subseteq S(w)\cup \{s\}$; this proves clause (2).

By the root-orbit definition [F2], $\rho (w)e_s\in \mathrm{\Phi }$; step 7.1 shows it lies in $V_{+}\setminus \{0\}$, so the sign split F3 gives $\rho (w)e_s\in {\mathrm{\Phi }}_{+}$. The root-length criterion F7 now gives $\mathrm{\ell }(ws)>\mathrm{\ell }(w)$, completing clause (1).

By the parabolic-support clause F6, the elements $u$ with $s\mathrm{\notin }S(u)$ are exactly $W_{S\setminus \{s\}}$. For any such $u$, $P(\mathrm{\ell }(u))$ supplies the expansion of (1), step 7.1 gives the support statement of (2) with $S(w)$ replaced by $S(u)$, and step 8.1 gives $\rho (u)e_s\in {\mathrm{\Phi }}_{+}$; hence $u\mapsto \rho (u)e_s$ maps $W_{S\setminus \{s\}}$ into ${\mathrm{\Phi }}_{+}$.