Euler-Mahonian statistics and descent bases for semigroup algebras

Abstract

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 show that the resulting graded quotient algebra has a basis where each element is indexed by colored permutations $(\pi,\epsilon)\in\mathbb{Z}_r\wr S_n$ and each element encodes the negative descent and negative major index statistics on $(\pi,\epsilon)$. This gives an algebraic interpretation of these statistics which was previously unknown. This basis of the $\mathbb{Z}_r\wr S_n$-quotients allows us to recover certain combinatorial identities involving Euler-Mahonian distributions of statistics.