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We consider quotients of the unit cube semigroup algebra by particular$\mathbb{Z}_r\wr S_n$-invariant ideals. Using Gr\"obner basis methods, we showthat the resulting graded quotient algebra has a basis where each element isindexed by colored permutations $(\pi,\epsilon)\in\mathbb{Z}_r\wr S_n$ and eachelement encodes the negative descent and negative major index statistics on$(\pi,\epsilon)$. This gives an algebraic interpretation of these statisticswhich was previously unknown. This basis of the $\mathbb{Z}_r\wr S_n$-quotientsallows us to recover certain combinatorial identities involving Euler-Mahoniandistributions of statistics.

Euler-Mahonian statistics and descent bases for semigroup algebras