Coxeter-matrix characterization of Auslander regular incidence algebras

Characterize Auslander regularity of the incidence algebra of a bounded finite poset by determining whether, for a linear-extension ordering of its simple modules, Auslander regularity is equivalent to the identity of the left triangular factor in a Bruhat decomposition of its Coxeter matrix.

Background

The paper gives a Coxeter-matrix characterization of distributive lattices and asks whether an analogous characterization can identify Auslander regular incidence algebras of posets. It notes that a general classification of Auslander regular posets is not known and that boundedness is equivalent to 1-Gorensteinness for incidence algebras.