Hecke Jucys-Murphy elements and q-contents

Let $R$ be a commutative ring, $q\in R^{\times }$, and $H=H_{R,q}(S_n)$ (The generic type-A Hecke algebra). For $1\le k\le n$ define $L_1:=0,L_k:=q^{-1}{T}_{(k-1,k)}+q^{-2}{T}_{(k-2,k)}+\cdots +q^{1-k}{T}_{(1,k)}\in H(k\ge 2),$ where $T_{(i,k)}$ is the standard basis element indexed by the transposition $(ik)$ (The standard basis of the generic type-A Hecke algebra). The $L_k$ are the Hecke Jucys-Murphy elements. They are fixed by the star anti-automorphism $T_w^{\ast }=T_{w^{-1}}$ (The star anti-automorphism of the type-A Hecke algebra): each transposition $(ik)$ is its own inverse, so $T_{(i,k)}^{\ast }=T_{(i,k)}$, and the coefficients $q^{i-k}$ are central, whence $L_k^{\ast }={\sum }_{i<k}{q}^{i-k}{T}_{(i,k)}=L_k$. They commute pairwise and every symmetric polynomial in $L_1,\dots ,L_n$ with coefficients in $R$ is central in $H$ (The Hecke Jucys-Murphy action is dominance triangular).

q-contents. The quantum integers $[m{]}_q$ for $m\ge 0$ are those of The $q$-integer, $q$-factorial and $q$-multinomial coefficients; extend the definition to negative indices by $[-m{]}_q:=-q^{-m}[m{]}_q$ for $m>0$, so that $(q-1)[m{]}_q=q^m-1$ for all $m\in \mathbb{Z}$. When $q-1$ is a unit this may be written $[m{]}_q=(q^m-1)\mathrm{/}(q-1)$; the polynomial definition requires no such division. Also $[m{]}_1=m$ for all $m\in \mathbb{Z}$, since $[-r{]}_1=-[r{]}_1=-r$. For a standard $\lambda$-tableau $t$ and $1\le k\le n$ let ${\mathrm{content}}_t(k)=c-r$ be the content of the node carrying $k$ (The content of a node and the content vector of a standard tableau) and define the q-content value $[{\mathrm{content}}_t(k){]}_q$. If $e=e(F,q)<\mathrm{\infty }$ for a field $F$ then $[m+e{]}_q=[m{]}_q$ for every $m\in \mathbb{Z}$ (The quantum characteristic of an invertible Hecke parameter), so the q-content value depends only on the residue of the content modulo $e$. At $q=1$ the identification $H_{R,1}(S_n)\cong R[S_n]$ with $T_w\mapsto w$ — the isomorphism given by the Coxeter presentation and the standard bases (The symmetric group has the Coxeter presentation, The standard basis of the generic type-A Hecke algebra) — sends $L_k$ to the ordinary Jucys-Murphy element $X_k={\sum }_{j<k}(jk)$ (The Jucys-Murphy elements of the symmetric group algebra), because each coefficient $q^{j-k}$ specializes to $1$, and $[{\mathrm{content}}_t(k){]}_1={\mathrm{content}}_t(k)$.