Generic Hecke algebra of a Coxeter system ($H_{\mathrm{gen}}(W,\{a_i\})$)

Over the polynomial parameter ring, the generic Hecke algebra has basis $(T_w{)}_{w\in W}$ and multiplicationThe parameters must agree for conjugate simple generators; equivalently, $(T_i-a_i)(T_i+1)=0$ together with the braid relations presents the algebra.

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