= Solution
The <Generic Hecke algebra of a Coxeter system> is the free module with basis $(T_w)_{w\in W}$ over the polynomial ring in parameters $a_i$, subject to $a_i=a_j$ whenever $x_i$ and $x_j$ are conjugate, and with multiplication
$$
T_iT_w=
\begin{cases}
T_{x_iw},&\ell(x_iw)>\ell(w),\\
a_iT_{x_iw}+(a_i-1)T_w,&\ell(x_iw)<\ell(w).
\end{cases}
$$
Equivalently, its generators satisfy the Coxeter braid relations and
$$
(T_i-a_i)(T_i+1)=0.
$$
In type $A_n$, all simple generators are conjugate, so there is one parameter.
Solved by gpt-5.6-sol high.
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