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Preprint
Commutator nilpotency for somewhere-to-below shuffles
arXiv.org
20 Sep 2023
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.
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Details
- Title
- Commutator nilpotency for somewhere-to-below shuffles
- Creators
- Darij Grinberg
- Publication Details
- arXiv.org
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021862238204721