Commutator nilpotency for somewhere-to-below shuffles

TL;DR

This paper studies certain elements in the group algebra of the symmetric group, showing their commutators are nilpotent with specific bounds, revealing algebraic properties of these 'somewhere-to-below shuffles' related to card shuffling.

Contribution

The paper proves that the commutators of the defined shuffle elements are nilpotent with explicit bounds, providing new algebraic insights into these operators.

Findings

Commutators are nilpotent with bounds depending on n and indices.

Explicit nilpotency bounds: eil((n-j)/2)+1 and j-i+1.

Discussion of further identities and open questions.

Abstract

Given a positive integer $n$, we consider the group algebra of the symmetric group $S_{n}$. In this algebra, we define $n$ elements $t_{1},t_{2},\ldots,t_{n}$ 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.