{c}
{wiki=Hecke_algebra#0-Hecke_algebra}

The 0-Hecke algebra is the <specialization of an algebra>[specialization] $a_i=0$ of a <Generic Hecke algebra of a Coxeter system>. Its generators obey $T_i^2=-T_i$.

0-Hecke algebra

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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 111 / 3 / a / Solution

{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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 Incoming links: Generic Hecke algebra of a Coxeter system