An Abstract Model for Bruhat Intervals
Résumé
We introduce two constructions, which we call extension and shift, that will construct new posets from old. These constructions are defined in terms of labellings of the elements of the poset by signs $+$, $-$, that axiomatize the ones induced by the left multiplication by a generator $s$ on an interval in the Bruhat ordering of a Coxeter group; in that case the label is $+$ when the length goes up, $-$ when it goes down. Starting from the one-element poset, these constructions yield a class of Eulerian posets containing, but not limited to, all intervals in (finitely generated) Coxeter groups. We explain how analogues of Kazhdan-Lusztig polynomials may be defined for such a poset in terms of a given construction of the poset. It is hoped that for a restricted set of constructions, the result will be independent of the choice of construction. We prove this for posets that do not contain dihedral subintervals of length more than two.