The one-sided cycle shuffles in the symmetric group algebra

Abstract

We study a family of shuffling operators on the symmetric group $S_n$, which includes the 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 (uniformly) random position further below. Rewritten in terms of the group algebra $\mathbb{R}[S_n]$, our shuffle corresponds to right multiplication by a linear combination of the elements \[t_i:=\text{cyc}_{i}+\text{cyc}_{i,i+1}+\text{cyc}_{i,i+1,i+2}+\cdots+\text{cyc}_{i,i+1,\ldots,n}\in \mathbb{R}[S_n]\] for all $i\in\{1,2,\ldots,n\}$ (where $\text{cyc}_{j_1,j_2,\ldots,j_p}$ stands for a $p$-cycle). We compute the eigenvalues of these shuffling operators and of all their linear combinations. In particular, we show that the eigenvalues of right multiplication by a linear combination $\lambda_1t_1+\lambda_2t_2+\cdots+\lambda_nt_n$ are the numbers $\lambda_1m_{I,1}+\lambda_2m_{I,2}+\cdots+\lambda_nm_{I,n}$, where $I$ ranges over the subsets of $\{1,2,\ldots,n-1\}$ that contain no two consecutive integers; here $m_{I,i}$ are certain integers. We compute the multiplicities of these eigenvalues and show that if they are all distinct, the shuffling operator is diagonalizable. To this purpose, we show that the operators of right multiplication by $t_1,t_2,\ldots,t_n$ on $\mathbb{R}[S_n]$ are simultaneously triangularizable (via a combinatorially defined basis). The results stated here over $\mathbb{R}$ for convenience are actually stated and proved over an arbitrary commutative ring $\mathbf{k}$. We finish by describing a strong stationary time for the random-to-below shuffle, which is the shuffle in which the card that moves below is selected uniformly at random, and we give the waiting time for this event to happen.