Abstract:  We consider quotients of the unit cube semigroup algebra by particular $\mathbb{Z} \wr S_n$ -invariant ideals. Using Groebner basis methods, we can show that the resulting graded quotient algebra has a basis with each element indexed by colored permutation and each element encodes descent and major index statistics of the colored permutation. We can use this basis to recover combinatorial identities. This talk is based on joint work with Ben Braun.