An abstract model for Bruhat intervals
Let \((W,S)\) be a Coxeter system. It has been shown by \textit{V. Deodhar} [Invent. Math. 39, 187--198 (1977; Zbl 0333.20041)] that the Bruhat ordering in \(W\) can be characterized in terms of some properties of its intervals. In the paper under review these properties are axiomatized and the abstract model for Bruhat intervals is studied. It is shown that the corresponding posets are always Eulerian and that chain complexes of their proper parts \(]\hat{0},\hat{1}[\) are triangulations of spheres. In the last section an attempt is made to define analogues of the Kazhdan-Lusztig polynomials in the abstract context. Although examples show that it is not possible in full generality, the author was able to reprove for a special class of adihedral posets a result of \textit{F.~Brenti} [Invent. Math. 118, No. 2, 371--394 (1994; Zbl 0836.20054)] stating that for an adihedral Bruhat interval the Kazhdan-Lusztig polynomial is equal to the Stanley \(g\)-polynomial of the dual interval.
scientific article; zbMATH DE number 1423288