Commutator nilpotency for somewhere-to-below shuffles

Abstract

Given a positive integer $n$, we consider the group algebra of the symmetric group $S_n$. In this algebra, we define $n$ elements $t_1,t_2,\dots ,t_n$ 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 ${\mathrm{cyc}}_{\ell ,\ell +1,\dots ,k}$ 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.

48 pages. Comments welcome! v2 simplifies some proofs and adds some more experimental data