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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
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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  • [BB] A. Björner, F. Brenti, Combinatorics of Coxeter groups, to appear.
  • [B1] F. Brenti, Combinatorial expansion of Kazhdan-Lusztig polynomials, J. London Math Soc. 55 (1997), 448-472. MR 99a:05143
  • [B2] F. Brenti, Lattice paths and Kazhdan-Lusztig polynomials, J. Amer. Math. Soc. 11 (1998), 229-259. MR 98h:20072
  • [De] V.V. Deodhar, On the root system of a Coxeter group, Eins. Math. 32 (1986), 611-630. MR 83j:20052a
  • [D1] Dyer, M., Reflection subgroups of Coxeter systems, J. of Alg. 135 (1991), 57-73. MR 91j:20100
  • [D2] Dyer, M., Hecke algebras and shellings of Bruhat intervals, Comp. Math. 89 (1993), 91-115. MR 95c:20053
  • [H] J.E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, 1990. MR 92h:20002

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Additional Information

Paola Cellini
Affiliation: Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Belzoni 7, 35131 Padova, Italy

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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