Geodesic
The absolute order of a permutation representation of a Coxeter group
Journal of Algebraic Combinatorics, Tome 39 (2014) no. 1, pp. 75-98.

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A permutation representation of a Coxeter group $W$ naturally defines an absolute order. This family of partial orders (which includes the absolute order on $W$) is introduced and studied in this paper. Conditions under which the associated rank generating polynomial divides the rank generating polynomial of the absolute order on $W$ are investigated when $W$ is finite. Several examples, including a symmetric group action on perfect matchings, are discussed. As an application, a well-behaved absolute order on the alternating subgroup of $W$ is defined.
Classification : 05E10, 05E15, 20F55
Keywords: Coxeter group, group action, absolute order, rank generating polynomial 
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     author = {Athanasiadis, Christos A. and Roichman, Yuval},
     title = {The absolute order of a permutation representation of a {Coxeter} group},
     journal = {Journal of Algebraic Combinatorics},
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     language = {en},
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Athanasiadis, Christos A.; Roichman, Yuval. The absolute order of a permutation representation of a Coxeter group. Journal of Algebraic Combinatorics, Tome 39 (2014) no. 1, pp. 75-98. http://geodesic.mathdoc.fr/item/JAC_2014__39_1_a6/