Quotient singularities in the Grothendieck ring of varieties

Let G G be a finite group, X X be a smooth complex projective variety with a faithful G G -action, and Y Y be a resolution of singularities of X / G X/G. Larsen and Lunts asked whether X / G − Y X/G-Y is divisible by A 1 A¹ in the Grothendieck ring of varieties. We show that the answer is negative if B G BG is not stably rational and affirmative if G G is abelian. The case when X = Z n X=Zⁿ for some smooth projective variety Z Z and G = S n G=Sₙ acts by permutation of the factors is of particular interest. We make progress on it by showing that Z n / S n − Z ⟨ n ⟩ / S n Zⁿ/Sₙ-Z n / Sₙ is divisible by A 1 A¹, where Z ⟨ n ⟩ Z n is Ulyanov’s polydiagonal compactification of the n n th configuration space of Z Z.