Euler–Mahonian statistics and descent bases for semigroup algebras

We consider quotients of the unit cube semigroup algebra by particular Z_r≀S_n-invariant ideals. Using Gröbner basis methods, we show that the resulting graded quotient algebra has a basis where each element is indexed by colored permutations (π,ϵ)∈Z_r≀S_n and each element encodes the negative descent and negative major index statistics on (π,ϵ). This gives an algebraic interpretation of these statistics that was previously unknown. This basis of the Z_r≀S_n-quotients allows us to recover certain combinatorial identities involving Euler–Mahonian distributions of statistics.