= Generic Hecke algebra of a Coxeter system
{c}
{title2=$H_{\mathrm{gen}}(W,\{a_i\})$}
{wiki=Iwahori–Hecke_algebra}
Over the polynomial parameter ring, the generic Hecke algebra has basis $(T_w)_{w\in W}$ and multiplication
$$
T_iT_w=
\begin{cases}
T_{x_iw},&\ell(x_iw)>\ell(w),\\
a_iT_{x_iw}+(a_i-1)T_w,&\ell(x_iw)<\ell(w).
\end{cases}
$$
The parameters must agree for conjugate simple generators; equivalently, $(T_i-a_i)(T_i+1)=0$ together with the braid relations presents the algebra.
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