Coxeter-matrix criterion for Auslander regularity of 1-Gorenstein algebras

Determine whether a 1-Gorenstein algebra of finite global dimension with an admissible ordering of its simple modules is Auslander regular whenever the left triangular factor in a Bruhat decomposition of its Coxeter matrix is the identity.

Background

The paper proves for Auslander regular algebras that, with an admissible ordering, the Coxeter matrix has a Bruhat decomposition whose left triangular factor is the identity. Question 5.1 asks whether, under the weaker assumptions of 1-Gorensteinness and finite global dimension, the converse implication holds. An example shows that the 1-Gorenstein assumption cannot simply be omitted.