A characterization of total reflection orders
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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.References
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Additional Information
- Paola Cellini
- Affiliation: Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Belzoni 7, 35131 Padova, Italy
- Email: cellini@math.unipd.it
- Received by editor(s): July 30, 1998
- Published electronically: October 27, 1999
- Communicated by: Ronald M. Solomon
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 1633-1639
- MSC (1991): Primary 20F55; Secondary 05E99
- DOI: https://doi.org/10.1090/S0002-9939-99-05234-X
- MathSciNet review: 1653429