A characterization of total reflection orders

Abstract:

Let $(W,S)$ be a Coxeter system with set of reflections $T$. It is known that if $\prec$ is a total reflection order for $W$, then, for each $s\in S$, $\{t\in T\mid t\prec s\}$ and its complement are stable under conjugation by $s$. Moreover the upper and lower $s$-conjugates of $\prec$ are still total reflection orders. For any total order $\prec$ on $T$, say that $\prec$ is stable if $\{t\in T\mid t\prec s\}$ is stable under conjugation by $s$ for each $s\in S$. We prove that if $\prec$ and all orders obtained from $\prec$ by successive lower or upper $S$-conjugations are stable, then $\prec$ is a total reflection order.