Remarks on the shuffle elements of Iwahori--Hecke algebras

Let $H_n(q)$ be the generic Iwahori–Hecke algebra associated to the symmetric group ${\mathfrak{S}}_n$ of degree $n$ and let ${\mathcal{Y}}_{n,i}$ be the sum of standard basis of $H_n(q)$ with Coxeter length $i$ for $1\le i\le \frac{n(n-1)}{2}$. We show that ${\mathcal{Y}}_{n,1}$ and ${\mathcal{Y}}_{n,2}$ are noncommutative whenever $n\ge 4$, which disproves Doikou’s Conjecture on the commutativity of the shuffle elements of $H_n(q)$ in (Nuclear Physics B 1029 (2026): 117532). We also show that the matrix of the operator of the left multiplication by ${\mathcal{Y}}_{n,i}$ in $H_n(q)$ is $q$-symmetric for all $i$.