Cyclotomic Murphy elements

The commuting property in [F1] makes each product in $u_{\lambda }$ a defined polynomial element. The presentation [F2] supplies the coefficient ring, unit parameters, and quotient in which these polynomials live. The row subgroup $S_{\lambda }$ is a product of finite symmetric groups on consecutive row intervals, so every $T_w$ in $x_{\lambda }$ is a defined finite Hecke element by [F3]. For standard $t$, the equality $t^{\lambda }\cdot d(t)=t$ determines $d(t)$ uniquely because the entry action is free; row-increasing entries mean that this representative is minimal in the row-stabilizer coset, by the parabolic description in [F3]. Thus all displayed expressions are elements of $H_{r,n}$.

It remains to verify the stated commutation. Let $T_i$ generate a row factor belonging to component $k$, so $i,i+1$ lie in that component's label interval. For a factor with index $c\le k$, [F5] gives $a_c<i$, so every $L_j$ in ${\prod }_{j\le a_c}(L_j-u_c)$ is distant from $T_i$ and commutes with it by [F1]. For a factor with $c>k$, [F5] gives $a_c\ge i+1$, so the factor contains both $(L_i-u_c)$ and $(L_{i+1}-u_c)$; the entire product over $j\le a_c$ is invariant under swapping $L_i,L_{i+1}$. The divided-difference term in [F1] is then zero, and [F1] gives commutation with $T_i$. Therefore $u_{\lambda }$ commutes with every generator of $S_{\lambda }$, hence with every $T_w$ in that group and with their sum $x_{\lambda }$.

Since the row subgroup is closed under inversion, $x_{\lambda }^{\ast }={\sum }_{w\in S_{\lambda }}{T}_{w^{-1}}=x_{\lambda }$. Also $u_{\lambda }^{\ast }=u_{\lambda }$ by [F4]. The commutation proved in step 2.1 gives $m_{\lambda }^{\ast }=(u_{\lambda }{x}_{\lambda }{)}^{\ast }=x_{\lambda }{u}_{\lambda }=u_{\lambda }{x}_{\lambda }=m_{\lambda }$. Applying the anti-involution to the defining formula for $m_{st}$ now yields $m_{st}^{\ast }=T_{d(t)}^{\ast }{m}_{\lambda }{T}_{d(s)}=m_{ts}$.