The one-sided cycle shuffles in the symmetric group algebra

We study an infinite family of shuffling operators on the symmetric group S n , which includes the well-studied top-to-random shuffle. The general shuffling scheme consists of removing one card at a time from the deck (according to some probability distribution) and re-inserting it at a position chosen uniformly at random among the positions below. Rewritten in terms of the group algebra ℝS n , our shuffle corresponds to right multiplication by a linear combination of the elements