Iwahori-Hecke algebra ( $H_{gen} (W, {a_{i}})$ )

Over the polynomial parameter ring, the generic Hecke algebra has basis

(T_{w})_{w \in W}

and multiplication

T_{i} T_{w} = {T_{x_{i} w}, a_{i} T_{x_{i} w} + (a_{i} - 1) T_{w}, ℓ (x_{i} w) > ℓ (w), ℓ (x_{i} w) < ℓ (w) .

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.

Table of contents
- 0-Hecke algebra Iwahori-Hecke algebra
- Iwahori-Hecke algebra of a BN-pair Iwahori-Hecke algebra
  - Hecke parameter of a BN-pair Iwahori-Hecke algebra of a BN-pair

0-Hecke 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}

.

Iwahori-Hecke algebra of a BN-pair ( $H_{k} (G, B)$ )

This is a finite-group realization of an Iwahori-Hecke algebra. 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.

Hecke parameter of a BN-pair ( $q_{i}$ )

For a representative

\overset{x}{˙}_{i} \in N

, the Hecke parameter is

q_{i} = [B : B \cap \overset{x}{˙}_{i} B \overset{x}{˙}_{i}^{- 1}] .

The Iwahori-Hecke algebra of a BN-pair is obtained from the generic algebra by

a_{i} \mapsto q_{i} \cdot 1_{k}

.

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The Iwahori–Hecke algebra is a mathematical structure that arises in the study of representation theory, particularly in the representation theory of the symmetric group and related algebraic objects, such as Coxeter groups and reductive algebraic groups. ### Definition The Iwahori–Hecke algebra, often denoted as \( \mathcal{H} \), is an algebra associated with a Coxeter group.

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