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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.

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    $\begingroup$ 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. $\endgroup$
    – Grant B.
    Nov 30, 2023 at 15:45

1 Answer 1

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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!

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