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 $\mathscr{Y}_{n,i}$ be the sum of standard basis of $H_n(q)$ with Coxeter length $i$ for $1\leq i\leq \frac{n(n-1)}{2}$. We show that $\mathscr{Y}_{n,1}$ and $\mathscr{Y}_{n,2}$ are noncommutative whenever $n\geq 4$, 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 $\mathscr{Y}_{n,i}$ in $H_n(q)$ is $q$-symmetric for all $i$.