I have a question about the proof of lemma 2.14 in Macdonald's paper The Poincaré Series of a Coxeter Group [1], where he used induction on $l(w)$ to prove that if $|E|=|R(w)|$, then $E=R(w)$. The induction step writes $w=w's_\beta$ for some $l(w')=l(w)-1$ and $s_\beta$ is a simple reflection. Macdonald argues that $\beta$ does not belong to $R(w')$, otherwise $w'$ would have a reduced expression ends with $s_\beta$, contradicts with $l(w's_\beta)>l(w')$.

I suspect that he made a mistake here: shouldn't we write $w=s_\beta w'$ here? Then by $l(w)>l(w')$ we can deduce that $w'^{-1}s_\beta$ is a positive root, therefore $\beta$ is not send to negative by $w'^{-1}$, hence $\beta\notin R(w')$.

Note: In Macdonald's paper, $R(w)$ is defined to be $\Phi^+\cap w\Phi^-$, it's the set of positive roots that are sent to negative by $w^{-1}$, not the set of positive roots that are sent to negative by $w$: $$\beta\in \Phi^+\cap w\Phi^-\Leftrightarrow \text{$\beta>0$ and $\beta=w\gamma$ for some $\gamma<0$} \Leftrightarrow \beta>0,\, w^{-1}\beta<0.$$

[1] *Macdonald, I. G.*, **The Poincaré series of a Coxeter group**, Math. Ann. 199, 161-174 (1972). ZBL0286.20062.