# Is this a typo in Macdonald's paper "The Poincaré Series of a Coxeter Group"?

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.

• Yes, it should be $w=s_\beta w'$. As further evidence, the formula $R(w)=s_\beta R(w')\cup \{s_\beta\}$ used in the paper only works for left multiplication. Nov 30, 2023 at 15:45

Posting my comment as an answer: yes, the paper should say $$w=s_\beta w'$$ instead of $$w's_\beta$$. Your reasoning is correct. Further evidence that there is a typo is given later in the proof of lemma 2.14, where it is used that $$R(w)=s_\beta R(w')\cup \{\beta\},$$ which is only true for left multiplication by $$s_\beta$$.
For reference, the set $$R(w)$$ is called the set of left inversions of $$w$$. Typos like this are called left/right issues and are somewhat common. Stay vigilant!