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Séminaire Lotharingien de Combinatoire, 78B.45 (2017), 12 pp.

# Thomas Gobet

# On Cycle Decompositions in Coxeter Groups

**Abstract.**
The aim of this note is to show that the cycle decomposition of
elements of the symmetric group admits a quite natural formulation in
the framework of dual Coxeter theory, allowing a generalization of it
to the family of so-called *parabolic quasi-Coxeter elements*
of Coxeter groups (in the symmetric group every element is a parabolic
quasi-Coxeter element). We show that such an element admits an
analogue of the cycle decomposition. Elements which are not in this
family still admit a generalized cycle decomposition, but it is not
unique in general.

Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.

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