Hecke algebra

A Hecke algebra is a deformation of the group algebra of a Coxeter group, obtained by deforming the quadratic relations for its simple generators while retaining the braid relations.

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 $T_i^2=-T_i$.

For a finite group $G$ with a BN-pair, the Iwahori-Hecke algebra over $k$ is the endomorphism algebra of the permutation module $k[G/B]$, up to the conventional opposite algebra. Its standard basis is indexed by the Bruhat double cosets.

For a representative ${\dot{x}}_i\in N$, the Hecke parameter isThe Iwahori-Hecke algebra of a BN-pair is obtained from the generic algebra by $a_i\mapsto q_i\cdot 1_k$.