Rational smoothness of varieties of representations for quivers of type $A$

In this paper, the authors study when the closure (in the Zariski topology) of orbits of representations of quivers of type $A$ are rationally smooth. This is done by considering the corresponding quantized enveloping algebra ${\mathbf{U}}$ and studying the action of the bar involution on PBW bases. Using Ringel's Hall algebra approach to quantized enveloping algebras and also Auslander-Reiten quivers, we can describe the commutation relations between root vectors. This way we get explicit formulae for the multiplication of an element of PBW bases adapted to a quiver with a root vector and also recursive formulae to study the bar involution on PBW bases. One of the consequences of our characterization is that if the orbit closure is rationally smooth, then it is smooth.