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A characterization of total reflection orders


Author: Paola Cellini
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
Published electronically: October 27, 1999
MathSciNet review: 1653429
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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.


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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

DOI: https://doi.org/10.1090/S0002-9939-99-05234-X
Keywords: Coxeter groups, total reflection orders
Received by editor(s): July 30, 1998
Published electronically: October 27, 1999
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 2000 American Mathematical Society

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