Given a positive integer $n$, we consider the group algebra of the symmetric group $S_{ n}$. In this algebra, we define $n$ elements $t_{ 1},t_{ 2},\ldots,t_{ n}$ by the formula \[ t_{ \ell}:=\operatorname*{ cyc}\nolimits_{ \ell}+\operatorname*{ cyc}\nolimits_{ \ell,\ell+1}+\operatorname*{ cyc}\nolimits_{ \ell,\ell+1,\ell+2}+\cdots+\operatorname*{ cyc}\nolimits_{ \ell,\ell+1,\ldots,n}, \] where $\operatorname*{ cyc}\nolimits_{ \ell,\ell+1,\ldots,k}$ denotes the cycle that sends $\ell\mapsto\ell+1\mapsto\ell+2\mapsto\cdots\mapsto k\mapsto\ell$. These $n$ elements are called the *somewhere-to-below shuffles* due to an interpretation as card-shuffling operators. In this paper, we show that their commutators $\left[ t_{ i},t_{ j}\right] =t_{ i}t_{ j}-t_{ j}t_{ i}$ are nilpotent, and specifically that \[ \left[ t_{ i},t_{ j}\right] ^{ \left\lceil \left( n-j\right) /2\right\rceil +1}=0\ \ \ \ \ \ \ \ \ \ \text{ for any }i,j\in\left\{ 1,2,\ldots,n\right\} \] and \[ \left[ t_{ i},t_{ j}\right] ^{ j-i+1}=0\ \ \ \ \ \ \ \ \ \ \text{ for any }1\leq i\leq j\leq n. \] We discuss some further identities and open questions.
Categories: math.CO math.RA